TITLE: How to find the posterior distribution
QUESTION [0 upvotes]: So suppose I have a coin that has a probability $\mu$ of landing on heads, and $1-\mu$ of landing on tails.
I am giving the prior distribution $\mu$ ~ Uniform[0,1], and my realization $D_1 = \{H,T\}$.
I was wondering how I could find the posterior distribution $p(\mu | D)$? I was thinking of applying Bayes rule which gives me $p(D_1 | \mu) p(\mu)$ over $p(D_1)$, but I have no idea how to find $p(D_1)$. After some calculation, I know that p(H) = 1/2, so I thought $p(D_1 | \mu) = 1/2 * 1/2 = 1/4$. Is this correct?

REPLY [0 votes]: I think I figured it out. Essentially, using the law of total probability I can expand P(D1) with an integral, and then since $\mu$ is uniformly distributed, $p(\mu)$ is 1. In addition, $p(D_1 | \mu)$ is $\mu * (1-\mu)$ since $\mu$ is the probability of heads.
Using those above facts, I can solve for $p(\mu | D)$. The idea I was missing was that $\mu$ is a random variable that isn't fixed, since this is Bayesian inference.


text:
:: Boolean Properties of Sets - Theorems
::  by Library Committee

environ

 vocabularies TARSKI, XBOOLE_0;
 notations TARSKI, XBOOLE_0;
 constructors TARSKI, XBOOLE_0;
 registrations XBOOLE_0;
 requirements BOOLE;
 definitions TARSKI, XBOOLE_0;
 equalities XBOOLE_0;
 expansions TARSKI, XBOOLE_0;
 theorems TARSKI, XBOOLE_0;

begin

reserve x,A,B,X,X9,Y,Y9,Z,V for set;

::$N Modus Barbara
theorem
  X c= Y & Y c= Z implies X c= Z;

theorem
  {} c= X;

theorem Th3:
  X c= {} implies X = {}
proof
  assume X c= {};
  hence X c= {} & {} c= X;
end;

theorem Th4:
  (X \/ Y) \/ Z = X \/ (Y \/ Z)
proof
  thus (X \/ Y) \/ Z c= X \/ (Y \/ Z)
  proof
    let x be object;
    assume x in (X \/ Y) \/ Z;
    then x in X \/ Y or x in Z by XBOOLE_0:def 3;
    then x in X or x in Y or x in Z by XBOOLE_0:def 3;
    then x in X or x in Y \/ Z by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in X \/ (Y \/ Z);
  then x in X or x in Y \/ Z by XBOOLE_0:def 3;
  then x in X or x in Y or x in Z by XBOOLE_0:def 3;
  then x in X \/ Y or x in Z by XBOOLE_0:def 3;
  hence thesis by XBOOLE_0:def 3;
end;

theorem
  (X \/ Y) \/ Z = (X \/ Z) \/ (Y \/ Z)
proof
  (X \/ Y) \/ Z = X \/ ((Z \/ Z) \/ Y) by Th4
    .= X \/ (Z \/ (Z \/ Y)) by Th4
    .= (X \/ Z) \/ (Y \/ Z) by Th4;
  hence thesis;
end;

theorem
  X \/ (X \/ Y) = X \/ Y
proof
  X \/ (X \/ Y) = (X \/ X) \/ Y by Th4
    .= X \/ Y;
  hence thesis;
end;

theorem Th7:
  X c= X \/ Y
by XBOOLE_0:def 3;

theorem Th8:
  X c= Z & Y c= Z implies X \/ Y c= Z
proof
  assume
A1: X c= Z & Y c= Z;
  let x be object;
  assume x in X \/ Y;
  then x in X or x in Y by XBOOLE_0:def 3;
  hence thesis by A1;
end;

theorem
  X c= Y implies X \/ Z c= Y \/ Z
proof
  assume
A1: X c= Y;
  let x be object;
  assume x in X \/ Z;
  then x in X or x in Z by XBOOLE_0:def 3;
  then x in Y or x in Z by A1;
  hence thesis by XBOOLE_0:def 3;
end;

theorem
  X c= Y implies X c= Z \/ Y
proof
  Y c= Z \/ Y by Th7;
  hence thesis;
end;

theorem
  X \/ Y c= Z implies X c= Z
proof
  X c= X \/ Y by Th7;
  hence thesis;
end;

theorem Th12:
  X c= Y implies X \/ Y = Y
proof
  assume
A1: X c= Y;
  thus X \/ Y c= Y
  proof
    let x be object;
    assume x in X \/ Y;
    then x in X or x in Y by XBOOLE_0:def 3;
    hence thesis by A1;
  end;
  let x be object;
  thus thesis by XBOOLE_0:def 3;
end;

theorem
  X c= Y & Z c= V implies X \/ Z c= Y \/ V
proof
  assume
A1: X c= Y;
  assume
A2: Z c= V;
  let x be object;
  assume x in X \/ Z;
  then x in X or x in Z by XBOOLE_0:def 3;
  then x in Y or x in V by A1,A2;
  hence thesis by XBOOLE_0:def 3;
end;

theorem
  (Y c= X & Z c= X & for V st Y c= V & Z c= V holds X c= V) implies X = Y \/ Z
proof
  assume that
A1: Y c= X & Z c= X and
A2: Y c= V & Z c= V implies X c= V;
  Y c= Y \/ Z & Z c= Y \/ Z by Th7;
  hence X c= Y \/ Z by A2;
  thus thesis by A1,Th8;
end;

theorem
  X \/ Y = {} implies X = {};

theorem Th16:
  (X /\ Y) /\ Z = X /\ (Y /\ Z)
proof
  thus (X /\ Y) /\ Z c= X /\ (Y /\ Z)
  proof
    let x be object;
    assume
A1: x in (X /\ Y) /\ Z;
    then
A2: x in Z by XBOOLE_0:def 4;
A3: x in X /\ Y by A1,XBOOLE_0:def 4;
    then
A4: x in X by XBOOLE_0:def 4;
    x in Y by A3,XBOOLE_0:def 4;
    then x in Y /\ Z by A2,XBOOLE_0:def 4;
    hence thesis by A4,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A5: x in X /\ (Y /\ Z);
  then
A6: x in Y /\ Z by XBOOLE_0:def 4;
  then
A7: x in Y by XBOOLE_0:def 4;
A8: x in Z by A6,XBOOLE_0:def 4;
  x in X by A5,XBOOLE_0:def 4;
  then x in X /\ Y by A7,XBOOLE_0:def 4;
  hence thesis by A8,XBOOLE_0:def 4;
end;

theorem Th17:
  X /\ Y c= X
by XBOOLE_0:def 4;

theorem
  X c= Y /\ Z implies X c= Y
proof
  Y /\ Z c= Y by Th17;
  hence thesis;
end;

theorem Th19:
  Z c= X & Z c= Y implies Z c= X /\ Y
proof
  assume
A1: Z c= X & Z c= Y;
  let x be object;
  assume x in Z;
  then x in X & x in Y by A1;
  hence thesis by XBOOLE_0:def 4;
end;

theorem
  (X c= Y & X c= Z & for V st V c= Y & V c= Z holds V c= X) implies X = Y /\ Z
proof
  assume that
A1: X c= Y & X c= Z and
A2: V c= Y & V c= Z implies V c= X;
  thus X c= Y /\ Z by A1,Th19;
  Y /\ Z c= Y & Y /\ Z c= Z implies Y /\ Z c= X by A2;
  hence thesis by Th17;
end;

theorem
  X /\ (X \/ Y) = X
proof
  thus X /\ (X \/ Y) c= X
  by XBOOLE_0:def 4;
  let x be object;
  assume
A1: x in X;
  then x in X \/ Y by XBOOLE_0:def 3;
  hence thesis by A1,XBOOLE_0:def 4;
end;

theorem Th22:
  X \/ (X /\ Y) = X
proof
  thus X \/ (X /\ Y) c= X
  proof
    let x be object;
    assume x in X \/ (X /\ Y);
    then x in X or x in X /\ Y by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 4;
  end;
  let x be object;
  thus thesis by XBOOLE_0:def 3;
end;

theorem Th23:
  X /\ (Y \/ Z) = X /\ Y \/ X /\ Z
proof
  thus X /\ (Y \/ Z) c= X /\ Y \/ X /\ Z
  proof
    let x be object;
    assume
A1: x in X /\ (Y \/ Z);
    then x in Y \/ Z by XBOOLE_0:def 4;
    then
A2: x in Y or x in Z by XBOOLE_0:def 3;
    x in X by A1,XBOOLE_0:def 4;
    then x in X /\ Y or x in X /\ Z by A2,XBOOLE_0:def 4;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in X /\ Y \/ X /\ Z;
  then x in X /\ Y or x in X /\ Z by XBOOLE_0:def 3;
  then
A3: x in X & x in Y or x in X & x in Z by XBOOLE_0:def 4;
  then x in Y \/ Z by XBOOLE_0:def 3;
  hence thesis by A3,XBOOLE_0:def 4;
end;

theorem Th24:
  X \/ Y /\ Z = (X \/ Y) /\ (X \/ Z)
proof
  thus X \/ Y /\ Z c= (X \/ Y) /\ (X \/ Z)
  proof
    let x be object;
    assume x in X \/ Y /\ Z;
    then x in X or x in Y /\ Z by XBOOLE_0:def 3;
    then x in X or x in Y & x in Z by XBOOLE_0:def 4;
    then x in X \/ Y & x in X \/ Z by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A1: x in (X \/ Y) /\ (X \/ Z);
  then x in X \/ Z by XBOOLE_0:def 4;
  then
A2: x in X or x in Z by XBOOLE_0:def 3;
  x in X \/ Y by A1,XBOOLE_0:def 4;
  then x in X or x in Y by XBOOLE_0:def 3;
  then x in X or x in Y /\ Z by A2,XBOOLE_0:def 4;
  hence thesis by XBOOLE_0:def 3;
end;

theorem
  (X /\ Y) \/ (Y /\ Z) \/ (Z /\ X) = (X \/ Y) /\ (Y \/ Z) /\ (Z \/ X)
proof
  thus X /\ Y \/ Y /\ Z \/ Z /\ X = (X /\ Y \/ Y /\ Z \/ Z) /\ (X /\ Y \/ Y /\
  Z \/ X) by Th24
    .= (X /\ Y \/ (Y /\ Z \/ Z)) /\ (X /\ Y \/ Y /\ Z \/ X) by Th4
    .= (X /\ Y \/ Z) /\ (X /\ Y \/ Y /\ Z \/ X) by Th22
    .= (X /\ Y \/ Z) /\ (X /\ Y \/ X \/ Y /\ Z) by Th4
    .= (X /\ Y \/ Z) /\ (X \/ Y /\ Z) by Th22
    .= (X \/ Z) /\ (Y \/ Z) /\ (X \/ Y /\ Z) by Th24
    .= (X \/ Z) /\ (Y \/ Z) /\ ((X \/ Y) /\ (X \/ Z)) by Th24
    .= (X \/ Y) /\ ((Y \/ Z) /\ (X \/ Z) /\ (X \/ Z)) by Th16
    .= (X \/ Y) /\ ((Y \/ Z) /\ ((X \/ Z) /\ (X \/ Z))) by Th16
    .= (X \/ Y) /\ (Y \/ Z) /\ (Z \/ X) by Th16;
end;

theorem Th26:
  X c= Y implies X /\ Z c= Y /\ Z
proof
  assume
A1: X c= Y;
  let x be object;
  assume
A2: x in X /\ Z;
  then x in X by XBOOLE_0:def 4;
  then
A3: x in Y by A1;
  x in Z by A2,XBOOLE_0:def 4;
  hence thesis by A3,XBOOLE_0:def 4;
end;

theorem
  X c= Y & Z c= V implies X /\ Z c= Y /\ V
proof
  assume that
A1: X c= Y and
A2: Z c= V;
  let x be object;
  assume
A3: x in X /\ Z;
  then x in Z by XBOOLE_0:def 4;
  then
A4: x in V by A2;
  x in X by A3,XBOOLE_0:def 4;
  then x in Y by A1;
  hence thesis by A4,XBOOLE_0:def 4;
end;

theorem Th28:
  X c= Y implies X /\ Y = X
proof
  assume
A1: X c= Y;
  thus X /\ Y c= X by Th17;
  let x be object;
  assume
A2: x in X;
  then x in Y by A1;
  hence thesis by A2,XBOOLE_0:def 4;
end;

theorem
  X /\ Y c= X \/ Z
proof
  X /\ Y c= X & X c= X \/ Z by Th7,Th17;
  hence thesis;
end;

theorem
  X c= Z implies X \/ Y /\ Z = (X \/ Y) /\ Z
proof
  assume
A1: X c= Z;
  thus X \/ Y /\ Z c= (X \/ Y) /\ Z
  proof
    let x be object;
    assume x in X \/ Y /\ Z;
    then
A2: x in X or x in Y /\ Z by XBOOLE_0:def 3;
    then x in X or x in Y & x in Z by XBOOLE_0:def 4;
    then
A3: x in (X \/ Y) by XBOOLE_0:def 3;
    x in Z by A1,A2,XBOOLE_0:def 4;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in (X \/ Y) /\ Z;
  then x in X \/ Y by XBOOLE_0:def 4;
  then
A5: x in X or x in Y by XBOOLE_0:def 3;
  x in Z by A4,XBOOLE_0:def 4;
  then x in X & x in Z or x in Y /\ Z by A5,XBOOLE_0:def 4;
  hence thesis by XBOOLE_0:def 3;
end;

theorem
  (X /\ Y) \/ (X /\ Z) c= Y \/ Z
proof
  now
    let x be object;
    assume x in (X /\ Y) \/ (X /\ Z);
    then x in (X /\ Y) or x in (X /\ Z) by XBOOLE_0:def 3;
    then x in X & x in Y or x in X & x in Z by XBOOLE_0:def 4;
    hence x in Y \/ Z by XBOOLE_0:def 3;
  end;
  hence thesis;
end;

Lm1: X \ Y = {} iff X c= Y
proof
  thus X \ Y = {} implies X c= Y
  by XBOOLE_0:def 5;
  assume
A1: X c= Y;
  now
    let x be object;
    x in X & not x in Y iff contradiction by A1;
    hence x in X \ Y iff x in {} by XBOOLE_0:def 5;
  end;
  hence thesis by TARSKI:2;
end;

theorem
  X \ Y = Y \ X implies X = Y
proof
  assume
A1: X \ Y = Y \ X;
  now
    let x be object;
    x in X & not x in Y iff x in Y \ X by A1,XBOOLE_0:def 5;
    hence x in X iff x in Y by XBOOLE_0:def 5;
  end;
  hence thesis by TARSKI:2;
end;

theorem Th33:
  X c= Y implies X \ Z c= Y \ Z
proof
  assume
A1: X c= Y;
  let x be object;
  assume
A2: x in X \ Z;
  then x in X by XBOOLE_0:def 5;
  then
A3: x in Y by A1;
  not x in Z by A2,XBOOLE_0:def 5;
  hence thesis by A3,XBOOLE_0:def 5;
end;

theorem Th34:
  X c= Y implies Z \ Y c= Z \ X
proof
  assume
A1: X c= Y;
  let x be object;
  assume
A2: x in Z \ Y;
  then not x in Y by XBOOLE_0:def 5;
  then
A3: not x in X by A1;
  x in Z by A2,XBOOLE_0:def 5;
  hence thesis by A3,XBOOLE_0:def 5;
end;

Lm2: X \ (Y /\ Z) = (X \ Y) \/ (X \ Z)
proof
  thus X \ (Y /\ Z) c= (X \ Y) \/ (X \ Z)
  proof
    let x be object;
    assume
A1: x in X \ (Y /\ Z);
    then not x in (Y /\ Z) by XBOOLE_0:def 5;
    then
A2: not x in Y or not x in Z by XBOOLE_0:def 4;
    x in X by A1,XBOOLE_0:def 5;
    then x in (X \ Y) or x in (X \ Z) by A2,XBOOLE_0:def 5;
    hence thesis by XBOOLE_0:def 3;
  end;
  (X \ Y) c= X \ (Y /\ Z) & X \ Z c= X \ (Y /\ Z) by Th17,Th34;
  hence thesis by Th8;
end;

theorem
  X c= Y & Z c= V implies X \ V c= Y \ Z
proof
  assume X c= Y & Z c= V;
  then X \ V c= Y \ V & Y \ V c= Y \ Z by Th33,Th34;
  hence thesis;
end;

theorem Th36:
  X \ Y c= X
by XBOOLE_0:def 5;

theorem
  X \ Y = {} iff X c= Y by Lm1;

theorem
  X c= Y \ X implies X = {}
proof
  assume
A1: X c= Y \ X;
  thus X c= {}
  proof
    let x be object;
    assume
A2: x in X;
    then x in Y \ X by A1;
    hence thesis by A2,XBOOLE_0:def 5;
  end;
  thus thesis;
end;

theorem Th39:
  X \/ (Y \ X) = X \/ Y
proof
  thus X \/ (Y \ X) c= X \/ Y
  proof
    let x be object;
    assume x in X \/ (Y \ X);
    then x in X or x in Y \ X by XBOOLE_0:def 3;
    then x in X or x in Y by XBOOLE_0:def 5;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in X \/ Y;
  then x in X or x in Y & not x in X by XBOOLE_0:def 3;
  then x in X or x in Y \ X by XBOOLE_0:def 5;
  hence thesis by XBOOLE_0:def 3;
end;

theorem
  (X \/ Y) \ Y = X \ Y
proof
  thus for x being object holds x in (X \/ Y) \ Y implies x in X \ Y
  proof let x be object;
    assume
A1: x in (X \/ Y) \ Y;
    then x in (X \/ Y) by XBOOLE_0:def 5;
    then
A2: x in X or x in Y by XBOOLE_0:def 3;
    not x in Y by A1,XBOOLE_0:def 5;
    hence thesis by A2,XBOOLE_0:def 5;
  end;
  thus for x being object holds x in X \ Y implies x in (X \/ Y) \ Y
  proof let x be object;
    assume
A3: x in X \ Y;
    then x in X or x in Y by XBOOLE_0:def 5;
    then
A4: x in (X \/ Y) by XBOOLE_0:def 3;
    not x in Y by A3,XBOOLE_0:def 5;
    hence thesis by A4,XBOOLE_0:def 5;
  end;
end;

theorem Th41:
  (X \ Y) \ Z = X \ (Y \/ Z)
proof
  thus for x being object holds x in (X \ Y) \ Z implies x in X \ (Y \/ Z)
  proof let x be object;
    assume
A1: x in (X \ Y) \ Z;
    then
A2: not x in Z by XBOOLE_0:def 5;
A3: x in (X \ Y) by A1,XBOOLE_0:def 5;
    then
A4: x in X by XBOOLE_0:def 5;
    not x in Y by A3,XBOOLE_0:def 5;
    then not x in (Y \/ Z) by A2,XBOOLE_0:def 3;
    hence thesis by A4,XBOOLE_0:def 5;
  end;
  thus for x being object holds x in X \ (Y \/ Z) implies x in (X \ Y) \ Z
  proof let x be object;
    assume
A5: x in X \ (Y \/ Z);
    then
A6: not x in (Y \/ Z) by XBOOLE_0:def 5;
    then
A7: not x in Y by XBOOLE_0:def 3;
A8: not x in Z by A6,XBOOLE_0:def 3;
    x in X by A5,XBOOLE_0:def 5;
    then x in (X \ Y) by A7,XBOOLE_0:def 5;
    hence thesis by A8,XBOOLE_0:def 5;
  end;
end;

theorem Th42:
  (X \/ Y) \ Z = (X \ Z) \/ (Y \ Z)
proof
  thus (X \/ Y) \ Z c= (X \ Z) \/ (Y \ Z)
  proof
    let x be object;
    assume
A1: x in (X \/ Y) \ Z;
    then x in (X \/ Y) by XBOOLE_0:def 5;
    then x in X & not x in Z or x in Y & not x in Z by A1,XBOOLE_0:def 3,def 5;
    then x in (X \ Z) or x in (Y \ Z) by XBOOLE_0:def 5;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in (( X \ Z) \/ (Y \ Z));
  then x in (X \ Z) or x in (Y \ Z) by XBOOLE_0:def 3;
  then
A2: x in X & not x in Z or x in Y & not x in Z by XBOOLE_0:def 5;
  then x in (X \/ Y) by XBOOLE_0:def 3;
  hence thesis by A2,XBOOLE_0:def 5;
end;

theorem
  X c= Y \/ Z implies X \ Y c= Z
proof
  assume
A1: X c= Y \/ Z;
  let x be object;
  assume
A2: x in X \ Y;
  then x in X by XBOOLE_0:def 5;
  then
A3: x in Y \/ Z by A1;
  not x in Y by A2,XBOOLE_0:def 5;
  hence thesis by A3,XBOOLE_0:def 3;
end;

theorem
  X \ Y c= Z implies X c= Y \/ Z
proof
  assume
A1: for x being object holds x in X \ Y implies x in Z;
  let x be object;
  assume x in X;
  then x in X \ Y or x in Y by XBOOLE_0:def 5;
  then x in Z or x in Y by A1;
  hence thesis by XBOOLE_0:def 3;
end;

theorem
  X c= Y implies Y = X \/ (Y \ X)
proof
  assume
A1: X c= Y;
  now
    let x be object;
    x in Y iff x in X or x in (Y \ X) by A1,XBOOLE_0:def 5;
    hence x in Y iff x in X \/ (Y \ X) by XBOOLE_0:def 3;
  end;
  hence thesis by TARSKI:2;
end;

theorem
  X \ (X \/ Y) = {} by Th7,Lm1;

theorem Th47:
  X \ X /\ Y = X \ Y
proof
  now
    let x be object;
    x in X & not x in X /\ Y iff x in X & not x in Y by XBOOLE_0:def 4;
    hence x in X \ X /\ Y iff x in X \ Y by XBOOLE_0:def 5;
  end;
  hence thesis by TARSKI:2;
end;

theorem
  X \ (X \ Y) = X /\ Y
proof
  thus for x being object holds x in X \ (X \ Y) implies x in X /\ Y
  proof let x be object;
    assume
A1: x in X \ (X \ Y);
    then not x in (X \ Y) by XBOOLE_0:def 5;
    then
A2: not x in X or x in Y by XBOOLE_0:def 5;
    x in X by A1,XBOOLE_0:def 5;
    hence thesis by A2,XBOOLE_0:def 4;
  end;
  thus for x being object holds x in X /\ Y implies x in X \ (X \ Y)
  proof let x be object;
    assume
A3: x in X /\ Y;
    then not x in X or x in Y by XBOOLE_0:def 4;
    then
A4: not x in (X \ Y) by XBOOLE_0:def 5;
    x in X by A3,XBOOLE_0:def 4;
    hence thesis by A4,XBOOLE_0:def 5;
  end;
end;

theorem Th49:
  X /\ (Y \ Z) = (X /\ Y) \ Z
proof
  now
    let x be object;
    x in X & x in Y & not x in Z iff x in X & x in Y & not x in Z;
    then x in X & x in (Y \ Z) iff x in (X /\ Y) & not x in Z by XBOOLE_0:def 4
,def 5;
    hence x in X /\ (Y \ Z) iff x in (X /\ Y) \ Z by XBOOLE_0:def 4,def 5;
  end;
  hence thesis by TARSKI:2;
end;

theorem Th50:
  X /\ (Y \ Z) = X /\ Y \ X /\ Z
proof
A1: X /\ Y c= X by Th17;
  X /\ Y \ X /\ Z = ((X /\ Y) \ X) \/ ((X /\ Y) \ Z) by Lm2
    .= {} \/ ((X /\ Y) \ Z) by A1,Lm1
    .= (X /\ Y) \ Z;
  hence thesis by Th49;
end;

theorem Th51:
  X /\ Y \/ (X \ Y) = X
proof
  thus X /\ Y \/ (X \ Y) c= X
  proof
    let x be object;
    assume x in X /\ Y \/ (X \ Y);
    then x in X /\ Y or x in (X \ Y) by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 4,def 5;
  end;
  let x be object;
  assume x in X;
  then x in X & x in Y or x in (X\Y) by XBOOLE_0:def 5;
  then x in X /\ Y or x in (X \ Y) by XBOOLE_0:def 4;
  hence thesis by XBOOLE_0:def 3;
end;

theorem Th52:
  X \ (Y \ Z) = (X \ Y) \/ X /\ Z
proof
  thus for x being object
    holds x in X \ (Y \ Z) implies x in (X \ Y) \/ X /\ Z
  proof let x be object;
    assume
A1: x in X \ (Y \ Z);
    then not x in (Y \ Z) by XBOOLE_0:def 5;
    then x in X & not x in Y or x in X & x in Z by A1,XBOOLE_0:def 5;
    then x in (X \ Y) or x in X /\ Z by XBOOLE_0:def 4,def 5;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
    assume x in (X \ Y) \/ X /\ Z;
    then x in (X \ Y) or x in X /\ Z by XBOOLE_0:def 3;
    then
A2: x in X & not x in Y or x in X & x in Z by XBOOLE_0:def 4,def 5;
    then not x in (Y \ Z) by XBOOLE_0:def 5;
    hence thesis by A2,XBOOLE_0:def 5;
end;

theorem
  X \ (Y \/ Z) = (X \ Y) /\ (X \ Z)
proof
  X \(Y \/ Z) c= X \ Y & X \ (Y \/ Z) c= X \ Z by Th7,Th34;
  hence X \ (Y \/ Z) c= (X \ Y) /\ (X \ Z) by Th19;
  let x be object;
  assume
A1: x in (X \ Y) /\ (X \ Z);
  then
A2: x in (X \ Y) by XBOOLE_0:def 4;
  then
A3: x in X by XBOOLE_0:def 5;
  x in (X \ Z) by A1,XBOOLE_0:def 4;
  then
A4: not x in Z by XBOOLE_0:def 5;
  not x in Y by A2,XBOOLE_0:def 5;
  then not x in (Y \/ Z) by A4,XBOOLE_0:def 3;
  hence thesis by A3,XBOOLE_0:def 5;
end;

theorem
  X \ (Y /\ Z) = (X \ Y) \/ (X \ Z) by Lm2;

theorem Th55:
  (X \/ Y) \ (X /\ Y) = (X \ Y) \/ (Y \ X)
proof
 for x being object holds
    x in (X \/ Y) \ (X /\ Y) iff x in (X \ Y) \/ (Y \ X)
  proof let x be object;
    thus x in (X \/ Y) \ (X /\ Y) implies x in (X \ Y) \/ (Y \ X)
    proof
      assume
A1:   x in (X \/ Y) \ (X /\ Y);
      then not x in (X /\ Y) by XBOOLE_0:def 5;
      then
A2:   not x in X or not x in Y by XBOOLE_0:def 4;
      x in (X \/ Y) by A1,XBOOLE_0:def 5;
      then x in X or x in Y by XBOOLE_0:def 3;
      then x in (X \ Y) or x in( Y \ X) by A2,XBOOLE_0:def 5;
      hence thesis by XBOOLE_0:def 3;
    end;
    assume x in (X \ Y) \/ (Y \ X);
    then x in (X \ Y) or x in (Y \ X) by XBOOLE_0:def 3;
    then x in X & not x in Y or x in Y & not x in X by XBOOLE_0:def 5;
    then ( not x in (X /\ Y))& x in (X \/ Y) by XBOOLE_0:def 3,def 4;
    hence thesis by XBOOLE_0:def 5;
  end;
  hence thesis by TARSKI:2;
end;

Lm3: X c= Y & Y c< Z implies X c< Z
proof
  assume that
A1: X c= Y and
A2: Y c< Z;
  Y c= Z by A2;
  hence X c= Z & X <> Z by A1,A2,XBOOLE_0:def 10;
end;

theorem
  X c< Y & Y c< Z implies X c< Z
by Lm3;

theorem
  not (X c< Y & Y c< X);

theorem
  X c< Y & Y c= Z implies X c< Z
proof
  assume that
A1: X c< Y and
A2: Y c= Z;
  X c= Y by A1;
  hence X c= Z & X <> Z by A1,A2,XBOOLE_0:def 10;
end;

theorem
  X c= Y & Y c< Z implies X c< Z by Lm3;

theorem Th60:
  X c= Y implies not Y c< X
proof
  assume X c= Y & Y c= X & X <> Y;
  hence contradiction;
end;

theorem
  X <> {} implies {} c< X
proof
  assume
A1: X <> {};
  thus {} c= X;
  thus thesis by A1;
end;

theorem
  not X c< {} by Th3;

::$N Modus Celarent
::$N Modus Darii
theorem Th63:
  X c= Y & Y misses Z implies X misses Z
by Th3,Th26;

theorem
  A c= X & B c= Y & X misses Y implies A misses B
proof
  assume that
A1: A c= X and
A2: B c= Y and
A3: X misses Y;
  A misses Y by A1,A3,Th63;
  hence thesis by A2,Th63;
end;

theorem
  X misses {};

theorem
  X meets X iff X <> {};

theorem
  X c= Y & X c= Z & Y misses Z implies X = {}
by Th3,Th19;

::$N Modus Darapti
theorem Th68:
  for A being non empty set st A c= Y & A c= Z holds Y meets Z
proof
  let A be non empty set;
  consider x being object such that
A1: x in A by XBOOLE_0:def 1;
  assume A c= Y & A c= Z;
  then x in Y & x in Z by A1;
  hence thesis by XBOOLE_0:3;
end;

theorem
  for A being non empty set st A c= Y holds A meets Y by Th68;

theorem Th70:
  X meets Y \/ Z iff X meets Y or X meets Z
proof
  thus X meets Y \/ Z implies X meets Y or X meets Z
  proof
    assume X meets Y \/ Z;
    then consider x being object such that
A1: x in X & x in Y \/ Z by XBOOLE_0:3;
    x in X & x in Y or x in X & x in Z by A1,XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:3;
  end;
A2: X meets Z implies X meets Y \/ Z
  proof
    assume X meets Z;
    then consider x being object such that
A3: x in X and
A4: x in Z by XBOOLE_0:3;
    x in Y \/ Z by A4,XBOOLE_0:def 3;
    hence thesis by A3,XBOOLE_0:3;
  end;
  X meets Y implies X meets Y \/ Z
  proof
    assume X meets Y;
    then consider x being object such that
A5: x in X and
A6: x in Y by XBOOLE_0:3;
    x in Y \/ Z by A6,XBOOLE_0:def 3;
    hence thesis by A5,XBOOLE_0:3;
  end;
  hence thesis by A2;
end;

theorem
  X \/ Y = Z \/ Y & X misses Y & Z misses Y implies X = Z
proof
  assume that
A1: X \/ Y = Z \/ Y and
A2: X /\ Y = {} and
A3: Z /\ Y = {};
  thus X c= Z
  proof
    let x be object such that
A4: x in X;
    X c= Z \/ Y by A1,Th7;
    then
A5: x in Z \/ Y by A4;
    not x in Y by A2,A4,XBOOLE_0:def 4;
    hence thesis by A5,XBOOLE_0:def 3;
  end;
  let x be object such that
A6: x in Z;
  Z c= X \/ Y by A1,Th7;
  then
A7: x in X \/ Y by A6;
  not x in Y by A3,A6,XBOOLE_0:def 4;
  hence thesis by A7,XBOOLE_0:def 3;
end;

theorem
  X9 \/ Y9 = X \/ Y & X misses X9 & Y misses Y9 implies X = Y9
proof
  assume
A1: X9 \/ Y9 = X \/ Y;
  assume X misses X9 & Y misses Y9;
  then
A2: X /\ X9 = {} & Y /\ Y9 = {};
  thus X = X /\ (X9 \/ Y9) by A1,Th7,Th28
    .= X /\ X9 \/ X /\ Y9 by Th23
    .= (X \/ Y) /\ Y9 by A2,Th23
    .= Y9 by A1,Th7,Th28;
end;

theorem
  X c= Y \/ Z & X misses Z implies X c= Y
proof
  assume that
A1: X c= Y \/ Z and
A2: X /\ Z = {};
  X /\ (Y \/ Z)= X by A1,Th28;
  then Y /\ X \/ {} = X by A2,Th23;
  hence thesis by Th17;
end;

theorem Th74:
  X meets Y /\ Z implies X meets Y
proof
  assume X meets Y /\ Z;
  then consider x being object such that
A1: x in X and
A2: x in Y /\ Z by XBOOLE_0:3;
  x in Y by A2,XBOOLE_0:def 4;
  hence thesis by A1,XBOOLE_0:3;
end;

theorem
  X meets Y implies X /\ Y meets Y
proof
  assume X meets Y;
  then consider x being object such that
A1: x in X and
A2: x in Y by XBOOLE_0:3;
  x in X /\ Y by A1,A2,XBOOLE_0:def 4;
  hence thesis by A2,XBOOLE_0:3;
end;

theorem
  Y misses Z implies X /\ Y misses X /\ Z
proof
  assume Y misses Z;
  then (X /\ Z) misses Y by Th74;
  hence thesis by Th74;
end;

theorem
  X meets Y & X c= Z implies X meets Y /\ Z
proof
  assume that
A1: X meets Y and
A2: X c= Z;
  now
    assume
A3: X /\ (Y /\ Z) = {};
    X /\ Y = (X /\ Z) /\ Y by A2,Th28
      .= {} by A3,Th16;
    hence contradiction by A1;
  end;
  hence thesis;
end;

theorem
  X misses Y implies X /\ (Y \/ Z) = X /\ Z
proof
  assume X misses Y;
  then X /\ Y = {};
  hence X /\ (Y \/ Z) = {} \/ X /\ Z by Th23
    .= X /\ Z;
end;

theorem Th79:
  X \ Y misses Y
proof
  not ex x being object st x in (X \ Y) /\ Y
  proof
    given x being object such that
A1: x in (X \ Y) /\ Y;
    x in X \ Y & x in Y by A1,XBOOLE_0:def 4;
    hence contradiction by XBOOLE_0:def 5;
  end;
  hence (X \ Y) /\ Y = {} by XBOOLE_0:def 1;
end;

theorem
  X misses Y implies X misses Y \ Z
proof
  assume
A1: X misses Y;
  assume X meets Y \ Z;
  then consider x being object such that
A2: x in X and
A3: x in Y \ Z by XBOOLE_0:3;
  x in Y by A3,XBOOLE_0:def 5;
  hence thesis by A1,A2,XBOOLE_0:3;
end;

theorem
  X misses Y \ Z implies Y misses X \ Z
proof
  X /\ (Y \ Z) = Y /\ X \ Z by Th49
    .= Y /\ (X \ Z) by Th49;
  hence thesis;
end;

theorem
  X \ Y misses Y \ X
proof
  assume X \ Y meets Y \ X;
  then consider x being object such that
A1: x in X \ Y and
A2: x in Y \ X by XBOOLE_0:3;
  x in X by A1,XBOOLE_0:def 5;
  hence thesis by A2,XBOOLE_0:def 5;
end;

theorem Th83:
  X misses Y iff X \ Y = X
proof
  thus X misses Y implies X \ Y = X
  proof
    assume
A1: X /\ Y = {};
    thus for x being object holds x in X \ Y implies x in X by XBOOLE_0:def 5;
    let x be object;
    not x in X /\ Y implies not x in X or not x in Y by XBOOLE_0:def 4;
    hence thesis by A1,XBOOLE_0:def 5;
  end;
  assume
A2: X \ Y = X;
  not ex x being object st x in X /\ Y
  proof
    given x being object such that
A3: x in X /\ Y;
    x in X & x in Y by A3,XBOOLE_0:def 4;
    hence contradiction by A2,XBOOLE_0:def 5;
  end;
  hence thesis by XBOOLE_0:4;
end;

theorem
  X meets Y & X misses Z implies X meets Y \ Z
proof
  assume that
A1: X meets Y and
A2: X misses Z;
  X /\ (Y \ Z) = X /\ Y \ X /\ Z by Th50
    .= X /\ Y \ {} by A2;
  hence X /\ (Y \ Z) <> {} by A1;
end;

theorem
  X c= Y implies X misses Z \ Y
proof
  assume
A1: X c= Y;
  thus X /\ (Z \ Y) = Z /\ X \ Y by Th49
    .= Z /\ (X \ Y) by Th49
    .= Z /\ {} by A1,Lm1
    .= {};
end;

theorem Th86:
  X c= Y & X misses Z implies X c= Y \ Z
proof
  assume
A1: X c= Y & X /\ Z = {};
  let x be object;
  assume x in X;
  then x in Y & not x in Z by A1,XBOOLE_0:def 4;
  hence thesis by XBOOLE_0:def 5;
end;

theorem
  Y misses Z implies (X \ Y) \/ Z = (X \/ Z) \ Y
proof
  assume
A1: Y misses Z;
  thus (X \/ Z) \ Y = (X \ Y) \/ (Z \ Y) by Th42
    .= (X \ Y) \/ Z by A1,Th83;
end;

theorem Th88:
  X misses Y implies (X \/ Y) \ Y = X
proof
  assume
A1: X misses Y;
  thus (X \/ Y) \ Y = (X \ Y) \/ (Y \ Y) by Th42
    .= (X \ Y) \/ {} by Lm1
    .= X by A1,Th83;
end;

theorem Th89:
  X /\ Y misses X \ Y
proof
  now
    let x be object;
    not (x in X & x in Y & not x in Y);
    hence not (x in X /\ Y & x in X \ Y) by XBOOLE_0:def 4,def 5;
  end;
  hence thesis by XBOOLE_0:3;
end;

theorem
  X \ (X /\ Y) misses Y
proof
  X \ (X /\ Y) = X \ Y by Th47;
  hence thesis by Th79;
end;

theorem
  (X \+\ Y) \+\ Z = X \+\ (Y \+\ Z)
proof
  set S1 = X \ (Y \/ Z), S2 = Y \ (X \/ Z), S3 = Z \ (X \/ Y), S4 = X /\ Y /\
  Z;
  thus (X \+\ Y) \+\ Z = (((X \ Y) \ Z) \/ ((Y \ X) \ Z)) \/ (Z \ ((X \ Y) \/
  (Y \ X))) by Th42
    .= (S1 \/ ((Y \ X) \ Z)) \/ (Z \ ((X \ Y) \/ (Y \ X))) by Th41
    .= (S1 \/ S2) \/ (Z \ ((X \ Y) \/ (Y \ X))) by Th41
    .= (S1 \/ S2) \/ (Z \ ((X \/ Y) \ (X /\ Y))) by Th55
    .= (S1 \/ S2) \/ (S4 \/ S3) by Th52
    .= (S1 \/ S2 \/ S4) \/ S3 by Th4
    .= (S1 \/ S4 \/ S2) \/ S3 by Th4
    .= (S1 \/ S4) \/ (S2 \/ S3) by Th4
    .= (S1 \/ X /\ (Y /\ Z)) \/ (S2 \/ S3) by Th16
    .= X \ ((Y \/ Z) \ (Y /\ Z)) \/ (S2 \/ S3) by Th52
    .= X \ ((Y \ Z) \/ (Z \ Y)) \/ (S2 \/ (Z \ (Y \/ X))) by Th55
    .= X \ ((Y \ Z) \/ (Z \ Y)) \/ ((Y \ (Z \/ X)) \/ (Z \ Y \ X)) by Th41
    .= X \ ((Y \ Z) \/ (Z \ Y)) \/ ((Y \ Z \ X) \/ (Z \ Y \ X)) by Th41
    .= X \+\ (Y \+\ Z) by Th42;
end;

theorem
  X \+\ X = {} by Lm1;

theorem Th93:
  X \/ Y = (X \+\ Y) \/ X /\ Y
proof
  thus X \/ Y = ((X \ Y) \/ X /\ Y) \/ Y by Th51
    .= (X \ Y) \/ (X /\ Y \/ Y) by Th4
    .= (X \ Y) \/ Y by Th22
    .= (X \ Y) \/ ((Y \ X) \/ (Y /\ X)) by Th51
    .= (X \+\ Y) \/ X /\ Y by Th4;
end;

Lm4: X /\ Y misses X \+\ Y
proof
  X /\ Y misses X \ Y & X /\ Y misses Y \ X by Th89;
  hence thesis by Th70;
end;

Lm5: X \+\ Y = (X \/ Y) \ X /\ Y
proof
  thus X \+\ Y = (X \ X /\ Y) \/ (Y \ X) by Th47
    .= (X \ X /\ Y) \/ (Y \ X /\ Y) by Th47
    .= (X \/ Y) \ X /\ Y by Th42;
end;

theorem
  X \/ Y = X \+\ Y \+\ X /\ Y
proof
  X /\ Y misses X \+\ Y by Lm4;
  then (X \+\ Y) \ X /\ Y = X \+\ Y & X /\ Y \ (X \+\ Y) = X /\ Y by Th83;
  hence thesis by Th93;
end;

theorem
  X /\ Y = X \+\ Y \+\ (X \/ Y)
proof
  X \+\ Y = (X \/ Y) \ X /\ Y by Lm5;
  then X \+\ Y c= X \/ Y by Th36;
  then
A1: (X \+\ Y) \ (X \/ Y) = {} by Lm1;
  X \/ Y = (X \+\ Y) \/ X /\ Y by Th93;
  hence thesis by A1,Lm4,Th88;
end;

theorem
  X \ Y c= X \+\ Y by Th7;

theorem
  X \ Y c= Z & Y \ X c= Z implies X \+\ Y c= Z by Th8;

theorem
  X \/ Y = X \+\ (Y \ X)
proof
A1: Y \ X \ X = Y \ (X \/ X) by Th41
    .= Y \ X;
  X \ (Y \ X) = (X \ Y) \/ X /\ X by Th52
    .= X by Th12,Th36;
  hence thesis by A1,Th39;
end;

theorem
  (X \+\ Y) \ Z = (X \ (Y \/ Z)) \/ (Y \ (X \/ Z))
proof
  thus (X \+\ Y) \ Z = (X \ Y \ Z) \/ (Y \ X \ Z) by Th42
    .= (X \ (Y \/ Z)) \/ (Y \ X \ Z) by Th41
    .= (X \ (Y \/ Z)) \/ (Y \ (X \/ Z)) by Th41;
end;

theorem
  X \ Y = X \+\ (X /\ Y)
proof
  X /\ Y c= X by Th17;
  then X /\ Y \ X = {} by Lm1;
  hence thesis by Th47;
end;

theorem
  X \+\ Y = (X \/ Y) \ X /\ Y by Lm5;

theorem
  X \ (Y \+\ Z) = X \ (Y \/ Z) \/ X /\ Y /\ Z
proof
  thus X \ (Y \+\ Z) = X \ ((Y \/ Z) \ Y /\ Z) by Lm5
    .= X \ (Y \/ Z) \/ X /\ (Y /\ Z) by Th52
    .= X \ (Y \/ Z) \/ X /\ Y /\ Z by Th16;
end;

theorem
  X /\ Y misses X \+\ Y by Lm4;

theorem
  X c< Y or X = Y or Y c< X iff X,Y are_c=-comparable;

begin :: Addenda

theorem
  for X, Y being set st X c< Y holds Y \ X <> {} by Th60,Lm1;

theorem Th106:
  X c= A \ B implies X c= A & X misses B
proof
  assume
A1: X c= A \ B;
  A \ B c= A by Th36;
  hence X c= A by A1;
  now
    let x be object;
    assume x in X;
    then x in A \ B by A1;
    hence not x in B by XBOOLE_0:def 5;
  end;
  hence thesis by XBOOLE_0:3;
end;

theorem
  X c= A \+\ B iff X c= A \/ B & X misses A /\ B
proof
  A \+\ B = (A \/ B) \ A /\ B by Lm5;
  hence thesis by Th86,Th106;
end;

theorem
  X c= A implies X /\ Y c= A
proof
  X /\ Y c= X by Th17;
  hence thesis;
end;

theorem Th109:
  X c= A implies X \ Y c= A
proof
  X \ Y c= X by Th36;
  hence thesis;
end;

theorem
  X c= A & Y c= A implies X \+\ Y c= A
proof
  assume X c= A & Y c= A;
  then X \ Y c= A & Y \ X c= A by Th109;
  hence thesis by Th8;
end;

theorem Th111:
  (X /\ Z) \ (Y /\ Z) = (X \ Y) /\ Z
proof
  thus (X /\ Z) \ (Y /\ Z) = ((X /\ Z) \ Y) \/ ((X /\ Z) \ Z) by Lm2
    .= ((X /\ Z) \ Y) \/ (X /\ (Z \ Z)) by Th49
    .= ((X /\ Z) \ Y) \/ (X /\ {}) by Lm1
    .= (X \ Y) /\ Z by Th49;
end;

theorem
  (X /\ Z) \+\ (Y /\ Z) = (X \+\ Y) /\ Z
proof
  thus (X /\ Z) \+\ (Y /\ Z) = ((X \ Y) /\ Z) \/ ((Y /\ Z) \ (X /\ Z)) by Th111
    .= ((X \ Y) /\ Z) \/ ((Y \ X) /\ Z) by Th111
    .= (X \+\ Y) /\ Z by Th23;
end;

theorem
  X \/ Y \/ Z \/ V = X \/ (Y \/ Z \/ V)
proof
  X \/ Y \/ Z \/ V = X \/ Y \/ (Z \/ V) by Th4
    .= X \/ (Y \/ (Z \/ V)) by Th4
    .= X \/ (Y \/ Z \/ V) by Th4;
  hence thesis;
end;

theorem
  for A,B,C,D being set st A misses D & B misses D & C misses D holds A
  \/ B \/ C misses D
proof
  let A,B,C,D be set;
  assume A misses D & B misses D;
  then
A1: A \/ B misses D by Th70;
  assume C misses D;
  hence thesis by A1,Th70;
end;

::$CT

theorem
  X /\ (Y /\ Z) = (X /\ Y) /\ (X /\ Z)
proof
  thus X /\ (Y /\ Z) = X /\ X /\ Y /\ Z by Th16
    .= X /\ (X /\ Y) /\ Z by Th16
    .= (X /\ Y) /\ (X /\ Z) by Th16;
end;

theorem
  for P,G,C being set st C c= G holds P \ C = (P \ G) \/ (P /\ (G \ C))
proof
  let P,G,C be set;
  assume C c= G;
  then
A1: P \ G c= P \ C by Th34;
  thus P \ C c= (P \ G) \/ (P /\ (G \ C))
  proof
    let x be object;
    assume x in P \ C;
    then x in P & not x in G or x in P & x in G & not x in C by XBOOLE_0:def 5;
    then x in P \ G or x in P & x in G \ C by XBOOLE_0:def 5;
    then x in P \ G or x in P /\ (G \ C) by XBOOLE_0:def 4;
    hence thesis by XBOOLE_0:def 3;
  end;
  P /\ (G \ C) = (P /\ G) \ C & (P /\ G) \ C c= P \ C by Th17,Th33,Th49;
  hence thesis by A1,Th8;
end;

subset_name: curated


text:
\begin{document}

\thispagestyle{empty}

\title[On the Transverse Invariant]{On the Transverse Invariant for Bindings of Open Books}
\author{David Shea Vela--Vick}
\address{Department of Mathematics \\ Columbia University}
\email{shea@math.columbia.edu}
\urladdr{\href{http://www.math.columbia.edu/~shea}{http://www.math.columbia.edu/\~{}shea}}

\date{\today}
\keywords{Legendrian knots, transverse knots, Heegaard Floer homology}
\subjclass[2000]{57M27; 57R58}
\maketitle

\begin{abstract}

Let $T \subset (Y,\xi)$ be a transverse knot which is the binding of some open book, $(T,\pi)$, for the ambient contact manifold $(Y,\xi)$.  In this paper, we show that the transverse invariant $\STH(T) \in \HFKH(-Y,K)$, defined in \cite{LOSS}, is nonvanishing for such transverse knots.  This is true regardless of whether or not $\xi$ is tight.  We also prove a vanishing theorem for the invariants $\SL$ and $\ST$.  As a corollary, we show that if $(T,\pi)$ is an open book with connected binding, then the complement of $T$ has no Giroux torsion.

\end{abstract}

\section{Introduction}
	\label{sec:intro}

In a recent paper by Lisca, Ozsv\'ath, Stipsicz, and Szab\'o \cite{LOSS}, the authors define invariants of null-homologous Legendrian and transverse knots.  These invariants live in the knot Floer homology groups of the ambient space with reversed orientation, and generalize the previously defined invariants of closed contact manifolds, $c(Y,\xi)$.  They have been useful in constructing new examples of knot types which are not transversally simple (see \cite{LOSS,OSt}), and play an important role in the classification of Legendrian and transverse twist knots (see \cite{ENV}).

In this paper, we investigate properties of these invariants for a certain important class of transverse knots.

\begin{theorem}\label{thm:binding}
Let $T \subset (Y,\xi)$ be a transverse knot which can be realized as the binding of an open book $(T,\pi)$ compatible with the contact structure $\xi$.  Then, the transverse invariant $\STH(T)$ is nonvanishing.
\end{theorem}

\begin{remark}
In \cite{LOSS}, it is shown that if $c(Y,\xi) \neq 0$, then $\ST(T) \neq 0$ for any transverse knot $T \subset (Y,\xi)$.  In \thmref{thm:binding}, no restrictions are made on the ambient contact structure $\xi$.  In particular, the theorem is true even when $\xi$ is overtwisted.  Moreover, the nonvanishing of the invariant $\STH$ implies the nonvanishing of the invariant $\ST$.
\end{remark}

Let $L$ be a null-homologous Legendrian knot in $(Y,\xi)$.  It is shown in \cite{LOSS} that the invariant $\SL(L)$ inside $\HFKM(-Y,L)$ remains unchanged under negative stabilization, and therefore yields an invariant of transverse knots.  If $T$ is a null-homologous transverse knot in $(Y,\xi)$ and $L$ is a Legendrian approximation of $T$, then $\ST(T) := \SL(L)$.  We will generally state results only in the Legendrian case, even though the same results are also true in the transverse case.

\begin{remark}
There is a natural map $\HFKM(-Y,L) \to \HFKH(-Y,L)$, induced by setting $U = 0$.  Under this map, the $\SL(L)$ is sent to $\SLH(L)$.  Therefore, if $\SLH(L)$ is nonzero, then $\SL(L)$ must also be nonzero.  Similarly, $\SL(L)$ vanishing implies that $\SLH(L)$ must also vanish.
\end{remark}

In addition to understanding when these invariants are nonzero, we are also interested in circumstances under which they vanish.  In \cite{LOSS}, it was shown that if the complement of a Legendrian knot contains an overtwisted disk, then the Legendrian invariant for that knot vanishes.  Here, we generalize this result by proving:

\begin{theorem}\label{thm:vanishing}
Let $L$ be a Legendrian knot in a contact manifold $(Y,\xi)$.  If the complement $Y-L$ contains a compact submanifold $N$ with convex boundary such that $c(N,\xi|_N) = 0$ in $SFH(-N,\Gamma)$, then the Legendrian invariant $\SL(L)$ vanishes.
\end{theorem}

Since $I$-invariant neighborhoods of convex overtwisted disks have vanishing contact invariant (Example 1 of \cite{HKM2}), \thmref{thm:vanishing} generalizes the vanishing theorem from \cite{LOSS}.

In \cite{GHV}, Ghiggini, Honda, and Van Horn--Morris show that a closed contact manifold with positive Giroux torsion has vanishing contact invariant.  They show this by proving that the contact element for a $2\pi$-torsion layer vanishes in sutured Floer homology.  Thus, as an immediate corollary to \thmref{thm:vanishing}, we have:

\begin{corollary}\label{cor:torsion}
Let $L$ be a Legendrian knot in a contact manifold $(Y,\xi)$.  If the complement $Y-L$ has positive Giroux torsion, then the Legendrian invariant $\SL(L)$ vanishes.
\end{corollary}

\begin{remark}
A similar result has been established for the weaker invariant $\SLH$ by Stipsicz and V\'ertesi \cite{StV} using slightly different arguments.
\end{remark}

\begin{remark}
\thmref{thm:binding} and Corollary \ref{cor:torsion} are both true in the transverse case as well.
\end{remark}

Combining the transverse version of Corollary \ref{cor:torsion} with \thmref{thm:binding}, we conclude the following interesting fact about complements of connected bindings:

\begin{theorem}\label{thm:torsion2}
Let $(T,\pi)$ be an open book with a single binding component.  Then the complement of $T$ has no Giroux torsion.
\end{theorem}

As Giroux torsion is presently the only known mechanism for a 3-manifold to admit more than a finite number of tight contact structures, it is important to understand the relationship between tight contact structures with positive Giroux torsion and the open books which support them.  Of course, Theorem~\ref{thm:torsion2} only applies to connected bindings of open books, leading one to conjecture that it should be true for arbitrary open book decompositions.  We prove this with Etnyre in \cite{EV} using different methods.

\begin{othertheorem}[Etnyre-Vela--Vick, \cite{EV}]\label{thm:torsion3}
Let $(B,\pi)$ be an open book for a contact manifold $(Y,\xi)$.  Then the complement of $B$ has no Giroux torsion.
\end{othertheorem}

This paper is organized as follows: in Section \ref{sec:background}, we briefly review some of the basic concepts in contact geometry and knot Floer homology.  Section \ref{sec:proof_of_main_thm} is devoted to proving \thmref{thm:binding}.  In Section \ref{sec:vanishing}, we conclude with a proof of \thmref{thm:vanishing}.

\section*{Acknowledgements}
\label{sec:acknowledgements}
I owe a tremendous debt of gratitude to my advisor, John Etnyre.  This problem arose in discussions with John.  His support and guidance over the years have been warmly received and much appreciated.  I also thank Clayton Shonkwiler for providing valuable comments on drafts of this paper.

\section{Background}
\label{sec:background}

\subsection{Contact Geometry Preliminaries}
\label{sub:contact_geometry_preliminaries}

Recall that a contact structure on an oriented 3-manifold is a plane field $\xi$ satisfying a certain nonintegrability condition.  We assume that our plane fields are cooriented, and that $\xi$ is given as the kernel of some global 1-form: $\xi = \mr{ker}(\alpha)$ with $\alpha(N_p) > 0$ for each oriented normal vector $N_p$ to $\xi_p$.  Such an $\alpha$ is called a {\it contact form for $\xi$}.  In this case, the nonintegrability condition is equivalent to the statement $\alpha \wedge d\alpha >0$.

A primary tool used in the study of contact manifolds has been Giroux's correspondence between contact structures on 3-manifolds and open book decompositions up to an equivalence called {\it positive stabilization}.  An {\it open book decomposition} of a contact 3-manifold $(Y,\xi)$ is a pair $(B,\pi)$, where $B$ is an oriented, fibered, transverse link and $\pi : (Y - B) \to S^1$ is a fibration of the complement of $B$ by oriented surfaces whose oriented boundary is $B$.

An open book is said to be {\it compatible} with a contact structure $\xi$ if, after an isotopy of the contact structure, there exists a contact form $\alpha$ for $\xi$ such that:
\begin{enumerate}
	\item $\alpha(v)>0$ for each (nonzero) oriented tangent vector $v$ to $B$, and
	\item $d\alpha$ restricts to a positive area form on each page of the open book.
\end{enumerate}

Given an open book decompositon of a 3-manifold $Y$, Thurston and Winkelnkemper \cite{TW} show how one can produce a compatible contact structure on $Y$.  Giroux proves in \cite{Gi} that two contact structures which are compatible with the same open book are, in fact, isotopic as contact structures.  Giroux also proves that two contact structures are isotopic if and only if they are compatible with open books which are related by a sequence of positive stabilizations.

\begin{definition}\label{def:Legendrian}
	A knot $L$ smoothly embedded in a contact manifold $(Y,\xi)$ is called {\it Legendrian} if $T_pL \subset \xi_p$ for all $p$ in $L$.
\end{definition}

\begin{definition}\label{def:transverse}
	An oriented knot $T$ smoothly embedded in a contact manifold $(Y,\xi)$ is called {\it transverse} if it always intersects the contact planes transversally with each intersection positive.
\end{definition}

We say that two Legendrian knots are {\it Legendrian isotopic} if they are isotopic through Legendrian knots; similarly, two transverse knots are {\it transversally isotopic} if they are isotopic through transverse knots.  Given a Legendrian knot $L$, one can produce a canonical transverse knot nearby to $L$, called the {\it transverse pushoff} of $L$.  On the other hand, given a transverse knot $T$, there are many nearby Legendrian knots, called {\it Legendrian approximations} of $T$.  Although there are infinitely many distinct Legendrian approximations of a given transverse knot, they are all related to one another by sequences of negative stabilizations.  These two constructions are inverses to one another, up to the ambiguity involved in choosing a Legendrian approximation of a given transverse knot (see \cite{EFM, EH}).

If $I$ is an invariant of Legendrian knots which remains unchanged under negative stabilization, then $I$ is also an invariant of transverse knots: if $T$ is a transverse knot and $L$ is a Legendrian approximation of $T$, define $I(T)$ to be equal to the invariant $I(L)$ of the Legendrian knot $L$.  This is how the authors define the transverse invariants $\ST(T)$ and $\STH(T)$ in \cite{LOSS}.

For more on open book decompositions and on Legendrian and transverse knots, we refer the reader to \cite{Et1,Co} and, respectively, \cite{Et2}.

\subsection{Heegaard Floer Preliminaries}
\label{sub:heegaard_floer_preliminaries}

This paper is primarily concerned with two versions of Heegaard Floer homology, which are invariants of (null-homologous) knots inside closed 3-manifolds.  These homologies, called knot Floer homology, are denoted $\HFKM(Y,K)$ and $\HFKH(Y,K)$.  In knot Floer homology, the basic input is a doubly-pointed Heegaard diagram; that is, a Heegaard diagram $(\Sigma,\alpha,\beta)$, together with two basepoints $w, z \in \Sigma - (\alpha \cup \beta)$, in the complement of the $\alpha$- and $\beta$-curves.  These diagrams are required to satisfy certain admissibility conditions which depend on the version of the theory which one is working.

Given a doubly pointed Heegaard diagram, one can produce a knot in the resulting 3-manifold.  To do this, connect $z$ to $w$ by an arc in the complement of the $\alpha$-curves, and $w$ to $z$ by an arc in the complement of the $\beta$-curves.  After depressing the interiors of these arcs into the $\beta$- and $\alpha$-handlebodies, respectively, the result is an oriented knot inside the closed 3-manifold specified by the Heegaard diagram $(\Sigma,\alpha,\beta)$.  Using a bit of elementary Morse theory, one can show that any knot in any closed 3-manifold can be represented by a doubly-pointed Heegaard diagram.

If the genus of $\Sigma$ is $g$, then the chain groups for $\HFKM(Y,K)$ are generated as a $\mathbb{Z}/2[U]$-module by the intersection points between the two $g$-dimensional subtori $\mathbb{T}_\alpha = \alpha_1 \times \dots \times \alpha_g$ and $\mathbb{T}_\beta = \beta_1 \times \dots \times \beta_g$ inside $\mr{Sym}^g(\Sigma)$.  Given a complex structure on $\Sigma$, $\mr{Sym}^g(\Sigma)$ inherits a natural complex structure from the projection $\times_g \Sigma \to \mr{Sym}^g(\Sigma)$.  The boundary map counts certain rigid holomorphic disks in $\mr{Sym}^g(\Sigma)$, with boundary lying on $\mathbb{T}_\alpha \cup \mathbb{T}_\beta$, connecting these intersection points:
\[
	\partial^- {\bf x} =
	\sum_{{\bf y} \in \mathbb{T}_\alpha \cap \mathbb{T}_\beta}
	\sum_{\substack{\phi \in \pi_2({\bf x}, {\bf y}), \,
	\mu(\phi)=1, \\
	n_z(\phi) = 0}}
	\# \widehat{\mfr{M}}(\phi) \cdot U^{n_w(\phi)} \cdot {\bf y}.
\]
Here $n_v(\phi)$ is equal to the algebraic number of times the disk $\phi$ intersects the subspace $\{v\} \times \mr{Sym}^{g-1}(\Sigma)$; $\pi_2({\bf x}, {\bf y})$ is the set of homotopy classes of disks connecting ${\bf x}$ to ${\bf y}$ with boundaries lying on $\mathbb{T}_\alpha$ and $\mathbb{T}_\beta$.

The chain groups for $\HFKH(Y,K)$ are generated as a $\mathbb{Z}/2$-vector space by the intersection points between $\mathbb{T}_\alpha$ and $\mathbb{T}_\beta$ in $\mr{Sym}^g(\Sigma)$.  In this case, the boundary map counts holomorphic disks in $\mr{Sym}^g(\Sigma)$, with boundaries lying on $\mathbb{T}_\alpha \cap \mathbb{T}_\beta$, missing both $z$ and $w$:
\[
	\widehat{\partial} {\bf x} =
	\sum_{{\bf y} \in \mathbb{T}_\alpha \cap \mathbb{T}_\beta}
	\sum_{\substack{\phi \in \pi_2({\bf x}, {\bf y}), \,
	\mu(\phi) = 1, \\
	n_z(\phi) = 0, \,
	n_w(\phi) = 0}}
	\# \widehat{\mfr{M}}(\phi) \cdot {\bf y}.
\]

For more information on Heegaard Floer homology and knot Floer homology, we refer the reader to \cite{OS1,Li} and \cite{OS3, Ra}, respectively.

\subsection{Invariants of Legendrian and Transverse Knots}
\label{sub:invariants_of_legendrian_and_transverse_knots}

Let $L$ be a Legendrian knot with knot type $K$, and let $T$ be a transverse knot in the same knot type.  In \cite{LOSS}, the authors define invariants $\SL(L)$ and $\ST(T)$ in $\HFKM(-Y,K)$ and $\SLH(L)$ and $\STH(T)$ in $\HFKH(-Y,K)$.  These invariants are constructed in a similar fashion to the contact invariants in \cite{HKM1,HKM2}.  Below we describe how to construct the invariant for a Legendrian knot.

Let $L \subset (Y,\xi)$ be a null-homologous Legendrian knot.  Consider an open book decomposition of $(Y,\xi)$ containing $L$ on a page $S$.  Choose a basis $\{a_0,\dots,a_n\}$ for $S$ (i.e a collection of disjoint, properly embedded arcs $\{a_0, \dots, a_n \}$ such that $S - \bigcup a_i$ is homeomorphic to a disk) with the property that $L$ intersects only the basis element $a_0$, and does so transversally in a single point.  Let $\{b_0,\dots,b_n\}$ be a collection of properly embedded arcs obtained from the $a_i$ by applying a small isotopy so that the endpoints of the arcs are isotoped according to the induced orientation on $\partial S$ and so that each $b_i$ intersects $a_i$ transversally in the single point $x_i$.  If $\phi: S \to S$ is the monodromy map representing the chosen open book decomposition, then our Heegaard diagram is given by
\[
	(\Sigma,\alpha,\beta) = (S_{1/2} \cup -S_1, (a_i \cup a_i), (b_i \cup \phi(b_i))).
\]

The first basepoint, $z$, is placed on the page $S_{1/2}$ in the complement of the thin strips of isotopy between the $a_i$ and $b_i$.  The second basepoint, $w$, is placed on the page $S_{1/2}$ inside the thin strip of isotopy between $a_0$ and $b_0$.  The two possible placements of $w$ correspond to the two possible orientations of $L$.

The Lengendrian invariant $\SL(L)$ is defined, up to isomorphism, to be the element $[{\bf x}] = [(x_0, \dots, x_n)]$ in $\HFKM(\Sigma,\beta,\alpha,w,z)$.  A picture of this construction in the case at hand is given in Figure~\ref{fig:basis}.  If $T$ is a transverse knot, the transverse invariant $\ST(T)$ is defined to be the Legendrian invariant of a Legendrian approximation of $T$.

One interesting property of these invariants is that they do not necessarily vanish for knots in an overtwisted contact manifolds; this is why we do not need to assume tightness in \thmref{thm:binding}.  Another property, which will be useful in Section \ref{sec:proof_of_main_thm}, is that these invariants are natural with respect to contact $(+1)$-surgeries.

\begin{othertheorem}[Ozsv\'ath-Stipsicz, \cite{OSt}]\label{thm:surgery}
Let $L \subset (Y,\xi)$ be a Legendrian knot.  If $(Y_B, \xi_B, L_B)$ is obtained from $(Y,\xi,L)$ by contact ($+1$)-surgery along a Legendrian knot $B$ in $(Y,\xi,L)$, then under the natural map
\[
	F_B : \HFKM(-Y,L) \to \HFKM(-Y_B,L_B),
\]
$\SL(L)$ is mapped to $\SL(L_B)$.
\end{othertheorem}

An immediate corollary to this fact is the following:

\begin{othercorollary}\label{cor:legsurgery}
Let $L \subset (Y,\xi)$ be a Legendrian knot, and suppose that $(Y_B, \xi_B, L_B)$ is obtained from $(Y,\xi,L)$ by Legendrian surgery along a Legendrian knot $B$ in $(Y,\xi,L)$.  If $\SL(L) \neq 0$ in $\HFKM(-Y,L)$, then $\SL(L_B) \neq 0$ in $\HFKM(-Y_B,L_B)$.
\end{othercorollary}

\begin{remark}
\thmref{thm:surgery} and Corollary \ref{cor:legsurgery} are also true for the invariant $\SLH(L)$ and for the invariants $\ST(T)$ and $\STH(T)$ in the case of a transverse knot.
\end{remark}

In addition, the invariant $\SL$ directly generalizes the original contact invariant $c(Y,\xi) \in \HFH(-Y)$ (see \cite{OS4}).  Under the natural map $\HFKM(-Y,L) \to \HFH(-Y)$ induced by setting $U = 1$, $\SL(L)$ maps to $c(Y,\xi)$, the contact invariant of the ambient contact manifold.

We encourage the interested reader to look at \cite{LOSS,OSt} to learn about other properties of these invariants.

\section{Proof of \thmref{thm:binding}}
\label{sec:proof_of_main_thm}

Let $T \subset (Y,\xi)$ be a transverse knot.  Recall that \thmref{thm:binding} states that if $T$ is the binding for some open book $(T,\pi)$ for $(Y,\xi)$, then the transverse invariant $\STH(T) \in \HFKH(-Y,T)$ is nonvanishing.

In this section, we prove \thmref{thm:binding} in three steps.  In Section \ref{sub:obtaining_the_pointed_diagram} we construct an open book on which a Legendrian approximation $L$ of the transverse knot $T$ sits.  Then we show in Section \ref{sub:admissibility} that the Heegaard diagram obtained in Section \ref{sub:obtaining_the_pointed_diagram} is weakly admissible.  Finally, in Section \ref{sub:computing_T(T)}, we prove the theorem in the special case where the monodromy map $\phi_n$ consists of a product of $n$ negative Dehn twists along a boundary-parallel curve.

An arbitrary monodromy map differs from some such $\phi_n$ by a sequence of positive Dehn twists, or Legendrian surgeries, along curves contained in pages of the open book.  By Corollary \ref{cor:legsurgery}, since the transverse invariant is nonvanishing for the monodromy maps $\phi_n$, it must also be nonvanishing for an arbitrary monodromy map.

\subsection{Obtaining the pointed diagram}
\label{sub:obtaining_the_pointed_diagram}

By hypothesis, $T$ is the binding of an open book $(T, \pi)$ for $(Y,\xi)$.  To compute the transverse invariant $\STH(T)$, we need to find a Legendrian approximation $L$ of $T$, realized as a curve on a page of an open book for $(Y,\xi)$.

\begin{figure}[htbp]
	\centering
	\subfigure[ ]{\label{fig:OpenBook1}
	\psfrag{a}{\hspace{-5px}$T$}
	\psfrag{b}{$\gamma$}
	\psfrag{c}{$S$}
	\includegraphics[scale=0.55]{OpenBook1.eps}}
	\subfigure[ ]{\label{fig:OpenBook2}
	\psfrag{a}{$\gamma$}
	\psfrag{b}{$c$}
	\psfrag{c}{$\tau$}
	\psfrag{d}{$T$}
	\psfrag{e}{$S^\prime$}
	\includegraphics[scale=0.55]{OpenBook2.eps}}
	\caption{ }
	\label{fig:OpenBook}
\end{figure}

In Figure~\ref{fig:OpenBook1}, we see a page of the open book $(T,\pi)$.  Here, $T$ appears as the binding $\partial S = T$.  Assuming the curve $\gamma$ could be realized as a Legendrian curve, it would be the natural choice for the Legendrian approximation $L$.  Unfortunately, since $\gamma$ is zero in the homology of the page, $\gamma$ cannot be made Legendrian on the page.

To fix this problem, stabilize the diagram.  The result of such a stabilization is illustrated in Figure~\ref{fig:OpenBook2}.  To see that this solves the problem, we prove the following lemma:

\begin{lemma}\label{lem:stabilization}
	The stabilization depicted in Figure~\ref{fig:OpenBook2} can be performed while fixing $T$ as the ``outer'' boundary component.
\end{lemma}

Assume the truth of Lemma \ref{lem:stabilization} for the moment.  Then the curve $\gamma$ depicted in Figure~\ref{fig:OpenBook2} can now be Legendrian realized, as it now represents a nonzero element in the homology of the page.  By construction, if we orient this Legendrian coherently with $T$, then $T$ is the transverse pushoff of $\gamma$.

\begin{proof}
Consider $S^3$ with its standard tight contact structure.  Let $(H_+,\pi_+)$ be the open book for $(S^3,\xi_{\mr{std}})$ whose binding consists of two perpendicular Hopf circles and whose pages consist of negative Hopf bands connecting these two curves.  In this case, each binding component is a transverse unknot with self-linking number equal to $-1$.

Let $T$ be a transverse knot contained in a contact manifold $(Y,\xi)$ and let $U$ be a transverse unknot in $(S^3,\xi|_{\mr{std}})$ with self-linking number equal to $-1$.  Observe that the complement of a standard neighborhood of a point contained in $U$ is itself a standard neighborhood of a point contained in a transverse knot.  Therefore, if we perform a transverse connected sum of $T$ with the transverse unknot of self-linking number equal to $-1$ in $(S^3, \xi_{\mr{std}})$, we do not change the transverse knot type of $T$.

\begin{figure}[htbp]
	\centering
	\subfigure[ ]{\label{fig:murasugi1}
		\hspace{-6pt}\includegraphics[scale=1.15]{Murasugi1.eps}}
	\hspace{2cm}
	\subfigure[ ]{\label{fig:murasugi2}
		\hspace{-6pt}\includegraphics[scale=1.15]{Murasugi2.eps}}
	\caption{ }
	\label{fig:Murasugi}
\end{figure}

Let $(B,\pi)$ be an open book with connected binding for a contact manifold $(Y,\xi)$.  Consider the contact manifold obtained from $(Y,\xi)$ by Murasugi summing the open book $(B,\pi)$ with the open book $(H_+,\pi_+)$ along bigon regions bounded by boundary-parallel arcs contained in pages of the respective open books.  The summing process is depicted in Figure~\ref{fig:Murasugi}.  Figure~\ref{fig:murasugi1} shows the open books before the Murasugi sum, while Figure~\ref{fig:murasugi2} shows the resulting open book after the sum.

\begin{figure}[htbp]
	\centering
	\subfigure[ ]{\label{fig:binding1}
		\hspace{-7pt}\includegraphics[scale=0.6]{binding1.eps}}
	\hspace{2cm}
	\subfigure[ ]{\label{fig:binding2}
		\hspace{-7pt}\includegraphics[scale=0.6]{binding2.eps}}
	\caption{ }
	\label{fig:binding}
\end{figure}

The Murasugi sum operation has the effect of performing a contact connected sum of $(Y,\xi)$ with $(S^3,\xi_{\mr{std}})$ and a transverse connected sum of the binding component $B$ with one of the binding components of $H_+$ (see \cite{To}).  Before and after pictures of this operation are shown in Figures \ref{fig:binding1} and \ref{fig:binding2}, respectively.  Since both of the binding components of the open book $(H_+,\pi_+)$ are transverse unknots with self-linking number equal to $-1$, this connected sum has no effect on the transverse knot type of the ``outer'' boundary component of the open book in Figure~\ref{fig:OpenBook}.
\end{proof}

Since the curve $\gamma$ can now be Legendrian realized and approximates $T$ as desired, we will denote $\gamma$ by $L$ from this point forward.  The new monodromy map $\phi^\prime : S^\prime \to S^\prime$ is equal to the old monodromy map, $\phi$, composed with one positive Dehn twist along the curve $c$ shown in Figure~\ref{fig:OpenBook2}.  For notational ease, we continue denoting the monodromy map by $\phi$, and the page by $S$.

\begin{figure}[htbp]
	\centering
	\psfrag{a}{$a_o$}
	\psfrag{b}{$w_1$}
	\psfrag{c}{$w_2$}
	\psfrag{d}{$z$}
	\psfrag{e}{$a_{2g}$}
	\psfrag{f}{$x_{2g}$}
	\psfrag{g}{$a_1$}
	\psfrag{h}{$x_1$}
	\psfrag{i}{$\dots$}
	\psfrag{j}{$L$}
	\psfrag{k}{$T$}
	\psfrag{l}{$x_0$}
        \includegraphics[scale=1.25]{basis.eps}
		\caption{}
		\label{fig:basis}
\end{figure}

We choose a basis for our surface whose local picture near the stabilization is depicted in Figure~\ref{fig:basis}.  There are two possible choices for the placement of the second basepoint $w$: $w_1$ and $w_2$.  In order for $L$ to be oriented coherently with $T$, we must choose $w = w_1$.

\subsection{Admissibility}
\label{sub:admissibility}

Our goal is to construct a weakly admissible, doubly-pointed Heegaard diagram from the open book described in Section \ref{sub:obtaining_the_pointed_diagram}.

Before we continue, let us discuss some notation.  We are concerned with open book decompositions whose pages are twice-punctured surfaces.  We picture a genus $g$ surface as a $4g$-sided polygon with certain boundary edges identified.  We choose the standard identification scheme, where the first and third edges are identified, as are the second and fourth edges, the fifth and seventh edges, and so on.  For convenience, we always assume that the first edge appears in the 12 o'clock position, at the top of each diagram.

\begin{figure}[htbp]
	\centering
	\psfrag{a}{$x_1$}
	\psfrag{b}{$x_2$}
	\psfrag{c}{$z$}
	\psfrag{d}{$x_0$}
	\psfrag{e}{$w$}
	\psfrag{f}{{\Large $S_{1/2}$}}
	\psfrag{g}{$a_{1,I}$}
	\psfrag{h}{$a_{2,I}$}
	\psfrag{i}{$a_{1,F}$}
	\psfrag{j}{$a_{2,F}$}
	\psfrag{k}{$a_0$}
	\psfrag{l}{$b_0$}
	    \includegraphics[scale=1.2]{contactpage.eps}
		\caption{}
		\label{fig:S_one_half}
\end{figure}

Our punctures are always situated so that one of the punctures is in the center of the polygon, with the other close by.  We choose our basis elements, $a_1, \dots, a_{2g}$, to be straight arcs emanating from the center of the polygon and passing out the corresponding edge.  If we were to forget about the identifications being made at the boundary, the basis element $a_i$ would break into two straight arcs emanating from the center of the diagram.  For ease of exposition in what follows, we label the first segment that we see as we move clockwise around the diagram $a_{i,I}$, and the second $a_{i,F}$, where the subscript $I$ stands for ``initial'', while the subscript $F$ stands for ``final''.

Up to isotopy, we may assume that the second boundary component of our surface lies (pictorially) in the region between the curves $a_{2g,F}$ and $a_{1,I}$, as shown in Figure~\ref{fig:S_one_half}.  The last basis element $a_0$ is a straight line segment connecting the two boundary components of the surface.

We have adopted the practice of Honda, Kazez and Mati\'c of placing surrogate basepoints throughout the diagram whenever it is convenient.  This signals that the local multiplicity of any domain contributing to the differential is zero in that region.

We have restricted our figures to the case where our page is a twice-punctured torus, and our monodromy map $\phi$ consists of two negative Dehn twists along the curve $\tau$ in Figure~\ref{fig:OpenBook2}.  The resulting doubly-pointed Heegaard diagram is shown in Figures \ref{fig:S_one_half} and \ref{fig:BigPicture}.

\begin{figure}[htbp]
	\centering
	\psfrag{a}{$a_{1,I}$}
	\psfrag{b}{$a_{2,I}$}
	\psfrag{c}{$a_{1,F}$}
	\psfrag{d}{$a_{2,F}$}
	\psfrag{e}{$(a_{1,I})$}
	\psfrag{f}{$(a_{2,I})$}
	\psfrag{g}{$(a_{1,F})$}
	\psfrag{h}{$(a_{2,F})$}
	\psfrag{i}{{\Large $-S_1$}}
	\psfrag{j}{$U$}
	\psfrag{k}{$R$}
	\psfrag{l}{$\gamma$}
	\psfrag{m}{$\delta$}
        \includegraphics[scale=1.65]{BigPicture.eps}
		\caption{}
		\label{fig:BigPicture}
\end{figure}

Figure~\ref{fig:S_one_half} shows the $S_{1/2}$ page of our open book, while Figure~\ref{fig:BigPicture} shows the $-S_1$ page (note the reversed orientation).  The invariant appears in Figure~\ref{fig:S_one_half} as the intersection point ${\bf x} = (x_0,x_1,x_2)$.

Consider the small region southeast of $x_0$ in Figure~\ref{fig:S_one_half}.  This region is equal to the region $R$ in Figure~\ref{fig:BigPicture}. Let $\gamma$ be the dashed arc connecting the region $R$ to the $z$-pointed region.  Denote by $(a_{i,*})$, the intersection point between $a_{i,*}$ and $\gamma$.

\begin{lemma}\label{lem:admissibility}
The doubly-pointed Heegaard diagram described above, and appearing in Figures \ref{fig:S_one_half} and \ref{fig:BigPicture}, is weakly admissible.
\end{lemma}

\begin{proof}
Let $P$ be a nontrivial periodic domain for the pointed Heegaard diagram $(\Sigma, \beta, \alpha, w)$.  Suppose $P$ has nonzero local multiplicity in the $z$-pointed region.  Without loss of generality, we assume that this multiplicity is $+1$.  In particular, the multiplicity of the region just above the point $(a_{2,F})$ in Figure~\ref{fig:BigPicture} is $+1$.  In order for the $w$-pointed region to have multiplicity zero, the $\alpha_0$- and $\beta_0$-curves must be contained in the boundary of the periodic domain $P$ and must appear with multiplicity $\pm 1$ (depending on the chosen orientations of $\alpha_0$ and $\beta_0$).  This forces the small region southeast of $x_0$ in Figure~\ref{fig:S_one_half} to have multiplicity $+2$.  Since this region is the same as the region $R$ in Figure~\ref{fig:BigPicture}, $R$ must also have multiplicity $+2$.

Consider the dashed arc $\gamma$ connecting $R$ to the $z$-pointed region.  In order for $P$ to exist, the multiplicities of the regions intersected by $\gamma$ must go from $+2$ in the region $R$ to $+1$ in the $z$-pointed region.

However, the curve $\gamma$ intersects each $\alpha$-curve (other than $\alpha_0$) in two points, each with opposite sign.  The boundary of a periodic domain must be a sum of full $\alpha$- and $\beta$-curves, so if the local multiplicity of $P$ increases (or decreases) by a factor of $n$ as $\gamma$ passes one of the intersection points, it must decrease (or increase) by that same factor as $\gamma$ passes the other intersection point.  Therefore, the net change in the local multiplicity of the periodic domain $P$ along $\gamma$ between the region $R$ and the $z$-pointed region is zero.  We have seen that the multiplicity of the region $R$ is $+2$, whereas the multiplicity of the $z$-pointed region was assumed to be $+1$.  From this contradiction, we conclude that $P$ must have local multiplicity zero in the $z$-pointed region.

Since each $\alpha$- and each $\beta$-curve bound the $z$-pointed region on either side, and since $P$ has local multiplicity zero in the $z$-pointed region, we conclude that $P$ must have both positive and negative coefficients.
\end{proof}

\subsection{Computing $\STH(T)$}
\label{sub:computing_T(T)}

Let $\phi \in \pi_2({\bf y}, {\bf y'})$ be a homotopy class with $n_z(\phi) = n_w(\phi) = 0$.  As is common in Heegaard Floer homology, we consider the ``shadow'' of $\phi$ on our Heegaard surface $\Sigma$.  This shadow is a sum of regions in the complement of the $\alpha$- and $\beta$-curves, and is denoted $D = \sum_i c_i D_i$.  If the homotopy class $\phi$ is to have a holomorphic representative $\mathbb{D}$, then each of the $c_i$ must be nonnegative.  In other words, $D$ is a positive domain and $\phi$ is a positive class.

Let ${\bf x} = (x_0, \dots, x_{2g})$; we show that the transverse invariant $\STH(T) = [(x_0, \dots, x_{2g})]$ is nonzero by proving that the generator ${\bf x}$ cannot appear in image of the Heegaard Floer differential.  This is accomplished by showing that the set of positive classes $\phi \in \pi_2({\bf y}, {\bf x})$ with $n_z(\phi) = n_w(\phi) = 0$ is empty for all ${\bf y} \neq {\bf x}$.

To draw a contradiction, we assume in what follows that $D$ is the domain of a positive class $\phi \in \pi_2({\bf y},{\bf x})$ for some generator ${\bf y} \neq {\bf x}$ with $n_z(\phi) = n_w(\phi) = 0$.

Suppose $R_1$ and $R_1$ are two adjacent regions in a Heegaard diagram (i.e. $R_1$ and $R_2$ share an edge), and $D$ is as above.  In general, the multiplicities of $R_1$ and $R_2$ can differ arbitrarily.  In our case, however, the multiplicities of any two adjacent regions can differ by at most one.  This is true because each of the $\alpha$- and $\beta$-curves in a Heegaard diagram coming from an open book decomposition bound the $z$-pointed region to either side.  Therefore, the boundary of any such region $D$ can never contain a full $\alpha$ or $\beta$ curve, and the multiplicities of $R_1$ and $R_2$ can differ by at most one.

Consider the region $R$ in Figure~\ref{fig:BigPicture}, and the curve $\gamma$ connecting $R$ to the $z$-pointed region.

\begin{lemma}\label{lem:nonnegative}
The net change in the local multiplicity of $D$ between the region $R$ and the $z$-pointed region along $\gamma$ is nonnegative.
\end{lemma}

\begin{proof}
The proof of Lemma \ref{lem:nonnegative} is similar to the proof of the admissibility lemma in Section \ref{sub:admissibility}.  	Recall that the curve $\gamma$ intersects each $\alpha_i$-curve in two points: $(a_{i,I})$ and $(a_{i,F})$.  We show that if the local multiplicity of the regions intersected by $\gamma$ decreases by a factor of $-1$ at the point $(a_{i,I})$, then there must be a corresponding increase in local multiplicity at the point $(a_{i,F})$.  Similarly, we show that if the local multiplicity of the regions intersected by $\gamma$ decrease by a factor of $-1$ at the point $(a_{i,F})$, then there must be a corresponding increase in local multiplicity at the point $(a_{i,I})$.

Observe that the local multiplicity along $\gamma$ cannot decrease as $\gamma$ passes over the point $(a_{1,I})$.  Since $D$ is the shadow of a positive class, if there is a decrease in local multiplicity at $(a_{1,I})$, a segment of the $\alpha_1$-curve between the intersection point $(a_{1,I})$ and $x_1$ must be contained in the boundary of the $D$.  Looking at the diagram in Figure~\ref{fig:BigPicture}, we see that any such arc has $z$-pointed region to the west, contradicting its existence.

In the genus one case, a similar argument shows that there can be no decrease in the local multiplicity of $D$ at the point $(a_{2,I})$.  So assume that either the genus of $S$ is greater than one, or that we are considering an intersection point $(a)$ beyond $(a_{2,I})$ along $\gamma$.

Suppose that $(a) = (a_{i,I})$, and that the local multiplicity along $\gamma$ decreases by a factor $-1$ as it passes over at the point $(a)$.  Then, up to orientation, the segment of the $\alpha_i$-curve beginning at the point $(a)$ and traveling away from the center of the diagram to the point $x_i$ is contained in the boundary of $D$.  This implies that the region just past the intersection point $(a_{i,F})$ along $\gamma$ gains a $+1$ boost in local multiplicity.  Therefore, the increase in the local multiplicity at the point $(a_{i,F})$ balances the decrease in the local multiplicity at the point $(a_{i,I})$.

Similarly, if $(a) = (a_{i,F})$ and if the local multiplicity of $D$ along $\gamma$ decreases by a factor of $-1$ as $\gamma$ passes over $(a)$, then, again up to orientation, the segment of the $\alpha_i$-curve beginning at the point $(a_{i,I})$ and traveling away from the center of the diagram to the point $x_i$ is contained in the boundary of $D$.  This implies that the region just past the intersection point $(a_{i,I})$ gains a $+1$ boost in local multiplicity.  Thus, the decrease in local multiplicity at the point $(a_{i,F})$ is balanced by the increase in local multiplicity at the point $(a_{i,I})$.

Since each decrease in the local multiplicity of $D$ along $\gamma$ is balanced by a corresponding increase in local multiplicity somewhere else along $\gamma$, we have that the net change in the local multiplicity of $D$ between the region $R$ and the $z$-pointed region along the curve $\gamma$ is nonnegative.
\end{proof}

Consider the region $U$ in Figure~\ref{fig:BigPicture}, and the curve $\delta$ connecting $U$ to the $z$-pointed region.  By an argument similar to the proof of Lemma \ref{lem:nonnegative}, we have the following:

\begin{lemma}\label{lem:nonnegative2}
The net change in the local multiplicity of $D$ between the region $U$ and the $z$-pointed region along $\delta$ is nonnegative.
\end{lemma}
\begin{flushright}
$\Box$
\end{flushright}

\begin{proof}[Proof of \thmref{thm:binding}]
There are two main cases to consider.

{\bf Case 1:}  Assume that the positive domain $D$ has nonzero local multiplicity in a region bordering the intersection point $x_0$.  In this case, the region $R$ in Figure~\ref{fig:BigPicture} has multiplicity $+1$.  By Lemma \ref{lem:nonnegative}, this implies that the multiplicity of the $z$-pointed region must be at least $+1$, a contradiction.

{\bf Case 2:}  Now suppose that $D$ has local multiplicity zero in all four of the regions bordering the point $x_0$.  In particular, this means that the multiplicity of the region $R$ is zero.  We investigate the possible configurations of $D$ near the center of Figure~\ref{fig:BigPicture}.

Suppose, for the moment, that all the regions bordered by the curve $\beta_0$ have zero multiplicity.  Then, near the center of Figure~\ref{fig:BigPicture}, the regions of $D$ with positive multiplicity are (locally) constrained to lie within the strip bounded by the darkened portions of the $\beta$-curves.

In order for this to be the case, the boundary of the domain must have veered off the $\alpha$-curves while still contained within this strip.  Therefore, all the $\alpha$-curves are ``used up'' close to the center of the diagram (i.e. by the time they first intersect a darkened $\beta$-curve).  This, in turn, forces the multiplicity of the region $U$ in Figure~\ref{fig:BigPicture} to be positive.

By Lemma \ref{lem:nonnegative2}, this implies that the multiplicity of the $z$-pointed region must be positive, a contradiction.  Therefore, in order for such a nontrivial positive domain to exist, at least one of the regions bordered by $\beta_0$ must have nonzero multiplicity.

Recall that in Case 2 we are assuming that our domain $D$ is constant near $x_0$.  This means that the curve $\beta_0$ cannot be appear in the boundary of $D$ with nonzero multiplicity, so at least one of the regions intersected by $\gamma$ must have positive multiplicity.  Let $R'$ be the first region along $\gamma$ with positive multiplicity, and let $(a)$ be the $(a_{i,*})$ immediately preceding $R'$.

If $(a) = (a_{i,F})$, then by an argument similar to the proof of Lemma \ref{lem:nonnegative}, it can be shown that the net change in multiplicity between the region $R'$ and the $z$-pointed region must be nonnegative.  The fact that $(a)$ is a final point ensures that there can be no decrease in multiplicity at the point $(a_{i,I})$ since, by the definition of $R'$, the regions to both sides of this point have multiplicity zero.

On the other hand, suppose $(a) = (a_{i,I})$.  An argument similar to that in Lemma \ref{lem:nonnegative} demonstrates that for each decrease in multiplicity, there is a corresponding increase in multiplicity, except possibly at the point $(a_{i,F})$.  If the multiplicity decreases at the point $(a_{i,F})$, then the segment of the $\alpha_i$-curve from $(a_{i,F})$ to $(a)$ must be contained in the boundary of the domain.  This then implies that the multiplicity of the region $U$ is at least one.

Again, by Lemma \ref{lem:nonnegative2}, this forces the multiplicity of the $z$-pointed region to be positive, a contradiction.
\end{proof}

This completes the proof of \thmref{thm:binding}.

\section{The vanishing theorem}
\label{sec:vanishing}

In this section, we prove \thmref{thm:vanishing}.  The proof in this case is similar to the proof of Theorem 4.5 in \cite{HKM2}. The key differences are that we must be careful to incorporate the Legendrian knot $L$ when choosing a Legendrian skeleton for the complement of the submanifold $N$, and that we must be cautious about the changes made to the diagram in the spinning process used to make the diagram strongly admissible.

\begin{proof}[Proof of \thmref{thm:vanishing}]
We begin by constructing a partial open book decomposition for the contact submanifold $(N,\xi|_N)$, which can be extended to an open book decomposition for all of $(Y,\xi)$.  Following \cite{HKM2}, we must show that the basis $\{a_1, \dots, a_r\}$ for the partial open book decomposition of $(N,\xi|_N)$ can be extended to a basis $\{a_1, \dots, a_r, a_0', a_1', \dots, a_s'\}$ for the extended open book decomposition of $(Y,\xi)$, where $L \cap (\bigcup a_i \cup \bigcup a_j') = L \cap a_0' = 1 \mr{ pt}$.\\

{\bf Claim:}  We may assume without loss of generality that the complement of $N$ is connected.

\begin{proof}[Proof of claim]
Let $(M,\xi)$ be a compact manifold with possibly nonempty boundary, and let $(M',\xi|_{M'})$ be a compact submanifold of $(M,\xi)$ with convex boundary.  In \cite{HKM3}, the authors show that the vanishing of the contact invariant for $(M',\xi|_{M'})$ implies the vanishing of the contact invariant for $(M,\xi)$.

Suppose the complement of $N$ is disconnected.  Then, since $c(N,\xi|_N) = 0$, the contact manifold obtained by gluing the components of $Y - N$ not containing $L$ to $N$ must also have vanishing contact invariant.  In particular, we may assume without loss of generality that $Y - N$ is connected.
\end{proof}

\begin{figure}[htbp]
	\centering
	\psfrag{a}{$L$}
	\psfrag{b}{$\nu(K)$}
	\psfrag{c}{\hspace{-10px}$\nu(K')$}
	\psfrag{d}{$N$}
	\psfrag{f}{$\partial N$}
		\includegraphics[scale=0.8]{skeleton.eps}
		\caption{}
		\label{fig:skeleton}
\end{figure}

Let $K$ be a Legendrian skeleton for $N$, and let $K'$ be an extension of the Legendrian knot $L$ to a Legendrian skeleton for $N' = Y - N$ (see Figure~\ref{fig:skeleton}).  Assume that the univalent vertices of $K$ and $K'$ in $\partial N$ do not intersect.

The Legendrian skeleton $K$ gives us a partial open book decomposition for $(N, \xi|_N)$.  Let $\nu(K)$ be a standard neighborhood of $K$ inside of $N$, and let $\nu(K')$ be a standard neighborhood of $K'$ inside of $Y - N$.  We can build an open book decomposition for all of $Y$ by considering the the handlebodies $(N' - \nu(K')) \cup \nu(K)$ and $\nu(K') \cup (N - \nu(K))$.  By construction, each of these handlebodies are disk decomposable.  A page $S$ of the open book for $(Y,\xi)$ is constructed from the page of the partial open book for $(N,\xi|_N)$ by repeatedly attaching 1-handles away from the portions of the open book coming from the boundary of $\nu(K)$.  This construction is depicted in Figure~\ref{fig:page}.

In Figure~\ref{fig:page}, the portion of the page of the open book coming from the boundary of $N$ is shown in black, and has its boundary lines thickened.  The portion of the page coming from the boundary of $\nu(K)$ is lightly colored (orange), and appears in the lower right portion of the figure.  Finally, the portion of the page coming from the extension of the open book to all of $Y$ is also lightly colored (green), and appears in the lower left corner of the figure.

Let $\{a_1,\dots,a_n\}$ be a basis for the partial open book coming from $(N,\xi|_N)$, and let $\phi$ be the corresponding partially defined monodromy map for this open book.  Consider a new partial open book, whose page is equal to $S$, and whose partially defined monodromy map is equal to $\phi$.  Because this new partial open book only differs from the partial open book coming from $(N,\xi|_N)$ by handle attachments away from $\partial \nu(K)$, the contact element for this new partial open book vanishes along with $c(N,\xi|_N)$.

Since $Y - N$ is connected, the basis $\{a_1, \dots, a_n \}$ can, after a suitable number of stabilizations, be extended to a basis for all of $Y$.

\begin{figure}[htbp]
	\centering
	\psfrag{a}{$\partial N$}
	\psfrag{b}{\hspace{-15px}$\partial \nu(K')$}
	\psfrag{c}{$\partial \nu(K)$}
	\psfrag{e}{\hspace{-2px}$a_0'$}
	\psfrag{g}{$L$}
	\psfrag{d}{$S$}
	\includegraphics[scale=0.85]{page.eps}
	\caption{}
	\label{fig:page}
\end{figure}

By construction, the new monodromy map $\phi'$ extends $\phi$, the monodromy map for $N$.  We can see our Legendrian $L$ on the page $S$.  The local picture around $L \subset S$ (shown in blue) must look like that in Figure~\ref{fig:page}.

As was observed in \cite{LOSS}, the ``spinning'' isotopies needed to make this Heegaard diagram strongly admissible can be performed on the portion of the Heegaard diagram coming from the page $S_1$.  This changes the monodromy map $\phi'$, but only within its isotopy class.

If we delete the $\alpha$- and $\beta$-curves coming from $\{a_0',a_1', \dots, a_s'\}$, then we are left with a diagram which is essentially equivalent to that coming from the partial open book $(S,\phi)$, but whose monodromy has been changed by an isotopy.  Since altering the monodromy map by an isotopy cannot change whether or not the contact element vanishes in sutured Floer homology, we know that the contact element corresponding to the partial open book $(S,\phi')$ vanishes.  That is, if ${\bf x} = (x_1, \dots, x_n)$, then there exist $c_i$ and ${\bf y}_i$ such that $\partial (\sum_i c_i {\bf y}_i) = {\bf x}$ in the sutured Floer homology of the manifold obtained from the partial open book $(S, \phi')$.

Let ${\bf x}' = (x_0', x_1', \dots, x_s')$; we claim that $\partial (\sum_i c_i ({\bf y}_i, {\bf x}')) = ({\bf x}, {\bf x}')$ in $\HFKM(-Y,L)$.  The intersection points coming from ${\bf x}'$ must map to themselves via the constant map.  This allows us to ignore the $\alpha$- and $\beta$-curves corresponding to these intersection points, leaving us with a diagram which is essentially equivalent to the partial open book $(S,\phi')$.
\end{proof}

\bibliographystyle{amsalpha}
\nocite{*}
\bibliography{TransInvt}

\end{document}

config: arxiv


TITLE: Adding an orientation to a given colored graph
QUESTION [1 upvotes]: I am writing my master thesis on double round robin tournament planning.
Part of my solution model is to add home/away-patterns to a fixed tournament plan. This plan is an edge coloring with the color indicating the round, and the edge indicating the match between two teams (vertices). In each round, every team must play, so the graph is a complete graph.
The orientation to be added indicates which of the two teams plays a home game.
The interesting constraint is, that no team is allowed to play home or away three matches in a row.
So, formally, I reckon the problem is something like: (and I am not an expert on defining problems)
Given a colored graph (with an ordered coloring), add an orientation to each edge, such that for each vertex, no three edges with adjacent colors are assigned the same orientation.
I have made an algorithm, which I have tested on a few instances and it has not failed to solve the problem yet. However, I cannot prove that it works. In addition, I cannot find any of these types of problems and thereby some proven ways to solve it.
My question is: Is this a standard problem, or is there an equivalent problem? And is there a nice way to solve it?
(And is 'an ordered coloring' just mumbo-jumbo-nonsense?)

REPLY [0 votes]: An ordered coloring is not mumbo-jumbo nonsense, but it is also not standard terminology. You could explain it, but personally I'd formalize your problem without colorings, as follows:

Our graph $G$ (or possibly multigraph, if two teams can play each other more than once) is the union of $k$ perfect matchings $M_1, M_2, \dots, M_k$ on the same set of $n$ vertices. (That is, each $M_i$ consists of $\frac n2$ edges that don't share any endpoints: it represents the games played in the $i^{\text{th}}$ round.)  Our goal is to orient the edges of $M_1, \dots, M_k$ such that for each $1 \le i \le k-2$ and  for each vertex $v$, both the out-degree and the in-degree of $v$ in $M_i \cup M_{i+1} \cup M_{i+2}$ are at least $1$.

In general, this topic is related to balanced orientations or almost-balanced orientations of graphs. See, for example, this question, where it is shown that any graph can be oriented so that the in-degree and out-degree of each vertex differ by at most $1$.
We could apply this to show that for each $i$, $M_i \cup M_{i+1} \cup M_{i+2}$ can be oriented in a way that satisfies this condition. However, doing this could land us in hot water: we also want $M_{i+1} \cup M_{i+2} \cup M_{i+3}$ to be oriented in this way, and the decisions we made for $M_{i+1}$ and $M_{i+2}$ without looking at $M_{i+3}$ might make this impossible.
Instead, here is a simple algorithm that is guaranteed to work: look at each of $M_1 \cup M_2$, $M_3 \cup M_4$, $M_5 \cup M_6$, and so on. Each of these $M_{2i-1} \cup M_{2i}$ is the union of two perfect matchings, so every vertex has degree $2$ in $M_{2i-1} \cup M_{2i}$. A graph in which every vertex has degree $2$ is either one big cycle or the union of multiple smaller cycles. In both cases, we can orient the edges around each cycle so that each vertex has in-degree $1$ and out-degree $1$.
This orientation is the home/away solution we're looking for! It has the following property: for every team, match $2i-1$ and match $2i$ involve playing one home game and one away game, in some order. Every block of three consecutive matches will contain an odd-numbered match followed by an even-numbered match, so it's impossible for a team to play three consecutive home games or three consecutive away games.


TITLE: General Existence and Uniqueness of ODE
QUESTION [6 upvotes]: I am trying to make sure that I understand the following question. As well as I am having a bit of trouble understand the partial solutions given.
The question is basically, what can we say about the following IVP ?
$$ y'=y^{1/3}$$ with initial value, $y(0)=0$ for $t \ge 0$
I know that the important theorem to be aware of is the theorem about unique solutions depending on continuity of the partial derivative w.r.t to y and existence depending on the continuity of f(t,y) itself.
So what I am seeing initially is that the partial derivative (w.r.t to y) is continuous everywhere expect at $y=0$. and  $f(t,y)$ itself is continuous for all values of $t$.
So does this Imply that we have a solution for all values of t, but we will only have a unique solution when $y \neq 0 $ ?
Solving, using the fact that we have a separable equation gives me,
$$y^{-1/3}dy=dt$$
Integrating both sides,
$$\frac{3}{2}y^{2/3}=t+c$$
$$y^{2/3}=\frac{2}{3}(t+c)$$
$$y^2=[(2/3)(t+c)]^{3}$$
$$y= \pm [(2/3)(t+c)]^{3/2}$$
From our initial conditions we can solve easily for c, to get that $c=0$
that is $y=\pm (2/3)t^{3/2}$
Now here is my first point of confusion, the solution says for both these answers, $t \ge 0$  I think I am just lost in it and even though I understood what the theorem stated, I don't understand its implications.
As well, from initial observation we know $y=\psi(t)=0$ is also a solution.
And the final answer is that $y= \lambda (t)= 0$ if $0 \le t \le t_{o}$ and $\pm[(2/3)(t-t_{o})]^{3/2}$ if $t \ge t_{o}$ Is the reason we have that the second part is only for $t \ge t_{0}$ just to avoid taking square root of negatives?
I am still confused about the final answer. I understand how we found that we could have $\pm [(2/3)(t+c)]^{3/2}$ and y=0 but what I am not understand is how from that we can obtain y=0 if $0 \le t \lt t_{o}$ and $\pm [(2/3)(t-t_{o})]^{3/2}$  if $t \ge t_{o}$? How was this done? It just directly stated this. Mainly, how is it that it went directly to saying
Can someone please help tie this together for me? Thanks a lot

REPLY [6 votes]: Your equation is of the form $y' = f(t,y)$ with $f(t,y) = y^{1/3}$.
Theorem (Short time existence): If $f(t,y)$ is continuous at $y_0$, then $y' = f(t,y)$, $y(t_0) = y_0$ has a solution on some interval containing $t_0$.
Your equation satisfies this.
Theorem (Short time uniqueness): If $f(t,y)$ is continuous at $y_0$ and $\frac{\partial f}{\partial y}$ exists and is continuous at $y_0$, then $y' = f(t,y)$, $y(t_0) = y_0$ has a unique solution on some interval containing $t_0$.
Your equation doesn't satisfy this for $y_0 = 0$, because $\frac{\partial}{\partial y} y^{1/3} = \frac{1}{3} y^{-2/3}$ does not exist at $y = 0$. So you don't have uniqueness of solutions starting with $y_0 = 0$.
If you start at some $y_0 \neq 0$, you do have uniqueness of solutions for a short time at least, in other words, on some time interval containing $t_0$. After a while, you might reach $y = 0$ on your solution, and uniqueness would fail. This theorem doesn't say anything in particular about long time uniqueness.
(Note: There are weaker conditions to put on $f$ in the theorems, but these are simple and work.)

Some intuition for what's going on:
If you sketch a slope field for $y' = y^{1/3}$, you'll note that there is a constant solution at $y = 0$. Often in such a case, there will be nearby solutions that asymptote to this solution. But in this case, $\frac{\partial f}{\partial y} = \frac{1}{3} y^{-2/3}$ (the rate at which the slope is changing) gets too large near $y = 0$, and solutions that start negative actually reach this constant solution.
Once that happens, uniqueness is shot: you can switch from the nearby solution $y = \left(\frac{2}{3} t\right)^{3/2}$ that has reached $y = 0$ to the $y = 0$ solution.
In fact, there are uncountably infinitely many solutions: you can stay on $y = 0$ for any distance and then start on one of the $\pm(2/3)^{3/2}(t+c)^{2/3}$ curves.

Note on weaker conditions:
As mentioned in the section above on intuition, the problem was related to the fact that the derivative $\frac{\partial f}{\partial y}$ becomes unbounded (arbitrarily large) near $y = 0$.
A weaker condition in the uniqueness theorem: $f(t,y)$ is Lipschitz in $y$. That is, there's some constant $C$ such that $|f(t,y_1) - f(t,y_2)| \leq C |y_1-y_2|$. (In fact you only need locally Lipschitz, which I'll leave to you to look up.) This really gets at the issue: $f$ should not be allowed to change too quickly, in the sense that secant slopes $\frac{|f(t,y_1) - f(t,y_2)|}{|y_1-y_2|}$ should be locally bounded. The existence and continuity of the derivative guarantees this, so it's a nice simpler-to-state condition to use in the theorem.
A wiki reference to the relevant theorem: Picard-Lindelöf theorem.

Note on why there are uncountably many solutions to $y' = y^{1/3}$, $y(t_0) = 0$:
$y = 0$ is a solution, as you noticed
$y = \pm[(2/3)(t-t_0)]^{3/2}$ is a solution, as you noticed
But so is
$y = 0$ for $t < T$ and $y = \pm[(2/3)(t-T)]^{3/2}$ for $t \geq T$ (for any $T \geq t_0$, so that $y(t_0) = 0$)
and so is
$y = 0$ for $t > T$ and $y = \pm[(2/3)(t-T)]^{3/2}$ for $t \leq T$ (for any $T \leq t_0$, so that $y(t_0) = 0$)
and so is
$y = \pm[(2/3)(t-S)]^{3/2}$ for $t \leq S$ and $y = 0$ for $S < y < T$ and $y = \pm[(2/3)(t-T)]^{3/2}$ for $t \geq T$ (for $S \leq t_0 \leq T$, so that $y(t_0) = 0$)
Why are these last three solutions as well?
The equation $y' = f(t,y)$ is an equation about the derivative of $y$. Both $y = 0$ and $y = \pm[(2/3)(t-T)]^{3/2}$ have the same derivative, namely zero, at $y = T$, so the piecewise function is differentiable with derivative zero there as well, so it must also be a solution to $y' = y^{1/3}$.
In fact, if you think carefully about the fact that the solutions are (on short time scale) unique away from $y = 0$, you'll see these are a complete list of the solutions.
Another comment on this:
If two solutions meet, they must have the same slope, because the equation $y' = f(t,y)$ determines the slope based on $t$ and $y$. So if two solutions meet (which precisely means you don't have uniqueness), you can always switch from one to the other as well.
The equation $y' = y^{1/3}$ is a very special situation where we have a whole line $y = 0$ of cases where uniqueness is violated, and this line is also itself a solution. In this case we get this uncountable family of solutions.


text:
\section{Discussion}

Gapped boundaries of topological models have been under scrutiny in the past years. Focusing on lattice Hamiltonian realisations of Dijkgraaf-Witten theory, a.k.a gauge models of topological phases, we studied gapped boundaries and their excitations in (2+1)d and (3+1)d. More specifically, the goal of this paper was two-fold: Apply the tube algebra approach to classify gapped boundary excitations and, using these results, elucidate the higher-category theoretical formalism relevant to describe gapped boundaries in (3+1)d.

As explained in detail in \cite{Bullivant:2019fmk}, local operators of lattice Hamiltonian realisations of Dijkgraaf-Witten theory can be conveniently expressed in terms of the partition function of the theory applied to so-called pinched interval cobordisms. We introduced a generalisation of the Dijkgraaf-Witten partition function, from which the gapped boundary Hamiltonian operators could be defined in analogy with the bulk Hamiltonian operators using the language of relative pinched interval cobordisms.  Given gapped boundaries labelled by subgroups of the input group and cochains compatible with the input cocycle, we applied the tube algebra approach in order to reveal the algebraic structure underlying two types of excitations: $(i)$ Point-like excitations at the interface of two gapped boundaries in (2+1)d, where the `tube' has the topology of $\mathbb I \times \mathbb I$, and $(ii)$ string-like (bulk) excitations terminating at point-like gapped boundary excitations, where the `tube' has the topology of $(\mathbb S^1 \times \mathbb I)\times \mathbb I$. Crucially, both tube algebras can be related via a lifting (or dimensional reduction) argument, and as such can be studied in parallel. This statement was formalised using the notion of relative groupoid algebra. When applied to the input group treated as a one-object groupoid, this notion yields the (2+1)d tube algebra, whereas it yields the (3+1)d tube algebra when applied to the loop groupoid of the group. We subsequently studied the representation theory of the (3+1)d tube algebra in full detail, which encompasses the (2+1)d one as a limiting case, deriving the irreducible representations as well as the corresponding recoupling theory.

In the second part of this manuscript, we reformulated the previous statements in category theoretical terms. In (2+1)d, the relevant notion to describe gapped boundaries and their excitations is the bicategory $\mathsf{MOD}(\mathsf{Vec}^{\alpha}_{G})$ of module categories over the category $\mathsf{Vec}^{\alpha}_{G}$ of group-graded vector spaces. In practice, a module category can be obtained as a category of modules over an algebra object in the input category. The bicategory of module categories above can then be shown to be equivalent to a bicategory of separable algebra objects, such that objects correspond to the gapped boundary conditions and morphisms to representations of a groupoid algebra isomorphic to the (2+1)d tube algebra. The identification with the tube algebra allowed us to elucidate the physical interpretation of the category theoretical notions at play. Mimicking this (2+1)d construction, we further defined a bicategory that encodes the string-like excitations terminating at point-like excitations on gapped boundaries and found that is was equivalent to a \emph{sub-bicategory} of the bicategory $\mathsf{MOD}(\mathsf{Vec}^{{\ssfT}(\pi)}_{\Lambda G})$ of modules categories over the category $\mathsf{Vec}^{{\ssfT}(\pi)}_{\Lambda G}$ of loop-groupoid-graded vector spaces. Comparing with the work of Kong et al. \cite{Kong:2019brm}, $\mathsf{MOD}(\mathsf{Vec}^{{\ssfT}(\pi)}_{\Lambda G})$ is equivalent to the higher categorical centre $\mathcal{Z}(\mathsf{2Vec}_G^\pi)$ of the category $\mathsf{2Vec}_G^\pi$ of $G$-graded 2-vector spaces, which is the input category of (3+1)d gauge models. In virtue of the physical interpretation inherited from the tube algebra approach, we thus suggested that $\mathcal{Z}(\mathsf{2Vec}_G^\pi)$ describes dyonic bulk string-like excitations whose end-points are pinned to the boundary of the spatial manifold. This is the higher-dimensional analogue of the well-known statement that bulk point-like excitations in (2+1)d are described by the centre $\mathcal Z(\mathsf{Vec}^\alpha_G)$ of  the input category.

The distinction between the gapped boundary string-like excitations we focused on, and the more general ones encoded in the centre $\mathcal Z(\mathsf{2Vec}^\pi_G)$ can be appreciated from an extended TQFT point of view. We should think of $\mathcal Z(\mathsf{2Vec}^\pi_G)$ as describing the object the extended 4-3-2-1 Dijkgraaf-Witten TQFT assigns to the circle. It follows from our analysis that such extended TQFT is more general than what gapped boundary conditions provide. Working out the details of this more general scenario will be the purpose of another paper.

The study carried out in this manuscript can be generalized in several ways. First of all, we could study gapped domains walls instead of gapped boundaries and consider string-like excitations that terminate at gapped domains walls point-like excitations. In (2+1)d, the so-called \emph{folding trick} can be used in order to map a gapped domain wall configuration to a gapped boundary one. It would certainly be interesting to consider how this generalizes in higher dimensions. Once this more general scenario is well-understood, we could then apply our results to so-called \emph{fracton} models, which were recently suggested in \cite{wen2020systematic,aasen2020topological, Wang:2020yma} to have an interpretation in terms of defect TQFTs. A related question would be to study invertible domain walls such as duality defects and derive the underlying mathematical structure in category theoretical terms.

Another follow-up work pertains to the relation between the string-like excitations as described by $\mathcal {Z}(\mathsf{2Vec}^\pi_G)$ and the loop-like excitations of the model. In a recent paper \cite{Bullivant:2019fmk}, the authors  showed that loop-like excitations and their statistics were captured by the category of modules over the so-called \emph{twisted quantum triple algebra}. This algebra can be expressed as the twisted groupoid algebra $\mathbb C[\Lambda^2G]^{\ssfT^2(\pi)}$ of the loop groupoid of the loop groupoid of $G$. In comparison, recall that the twisted quantum double is isomorphic to $\mathbb C[\Lambda G]^{\ssfT(\alpha)}$ in this language. This groupoid algebra was shown by the authors to be isomorphic to the tube algebra associated with the manifold $\mathbb T^2 \times \mathbb I$, a local neighbourhood of a loop-like object being a solid torus. Intuitively, we may expect loop-like excitations to descend from the string-like ones via a tracing mechanism. This can be formalized using the notion of \emph{categorical trace}, building upon the fact that it maps a module category over $\mathsf{Vec}_G$ to a module over $\mathbb C[\Lambda G]$ \cite{bartlett2009unitary, ganter2006representation}. Another way to establish the connection between string-like and loop-like excitations consists in first realising that, as braided monoidal categories, we have the equivalences $\mathcal Z(\mathsf{Vec}_{\Lambda G}^{\ssfT(\pi)}) \simeq \mathsf{Mod}(\mathbb C[\Lambda^2 G]^{\ssfT^2(\pi)})$ and $\mathcal Z(\mathsf{Vec}_{\Lambda G}^{\ssfT(\pi)}) \simeq \mathsf{Dim}(\mathsf{MOD}(\mathsf{Vec}^{\ssfT(\pi)}_{\Lambda G}))$, where $\mathsf{Dim}$ denotes the \emph{dimension of a bicategory} \cite{bartlett2009unitary,Baez:1995xq} obtained via an appropriate categorification of the dimension of a vector space. The details of this correspondence will be presented in a forthcoming paper \cite{Bullivant:2021pkd}.


TITLE: continuous and line spectra during beta and alpha particle emission
QUESTION [4 upvotes]: What is the reason behind alpha particle giving line spectra and beta particle giving continuous spectra?
(To notify this question had already been asked on this site. Yet, I am not quite clear with its explanation. I wish someone could answer it in simple language without much of mathematics involved.)

REPLY [1 votes]: I am assuming you know the model of the nucleus being surrounded by electrons, moving up and down between energy levels over time.
In an slightly similiar way, the neutrons and protons in the nucleus can be visualised as a set of shells, based again around energy levels.   A simple alpha decay will release  a set amount of energy. This is why you will get a discrete spectrum, as CuriousOne says, it's a two body problem, just the nucleus and the alpha particle.
When a beta particle leaves the nucleus,  you have 3 bodies involved, the nucleus, an electron (or positron) and a neutrino. Now there are many more ways the momentum and energy can be shared out, between the 3 bodies,  giving rise to a continuous spectrum, due to the laws of conservation of energy and momentum.