Calculation of quandle cocycle invariants via marked …gt.postech.ac.kr/satellite2014/static/slides/jieon_kim.pdf · Jieon Kim (Jointly with S. Kamada and S. Y. Lee) (PNU)Calculation

Calculation of quandle cocycle invariants viamarked graph diagrams

Jieon Kim(Jointly with S. Kamada and S. Y. Lee)

Pusan National University, Busan, Korea

August 25, 2014

Knots and Low Dimensional Manifolds

Jieon Kim (Jointly with S. Kamada and S. Y. Lee) (PNU)Calculation of quandle cocycle invariants via marked graph diagramsAugust 25, 2014 1 / 25

Contents

1 Surface-links

2 Quandle coloring invariants of oriented surface-links

3 Quandle cocycle invariants of oriented surface-links


Contents

1 Surface-links




Surface-links• A surface-link is a closed surface smoothly embedded in R4.• If a surface-link is oriented, then we call it an oriented

surface-link.• A marked graph diagram

Γ

Γ+

>>

>

>

>

>

>

>

Γ-


A marked graph diagram Γ is said to be admissible if bothresolutions Γ+ and Γ− are link diagrams of trivial links.

Theorem (Kearton-Kurlin, Swenton, Yoshikawa)(1) Let L be a surface-link. Then there is an admissible

marked graph diagram Γ s.t. L is presented by Γ.(2) Let Γ be an admissible marked graph diagram. Then there

is a surface-link L s.t. L is presented by Γ.


0

Γ

-1

1> >

>>

Γ-

Γ+

Γ


Contents

1 Surface-links




QuandlesA quandle is a set X with a binary operation ∗ : X×X → Xsatisfying

(Q1) For any x ∈ X, x∗ x = x.(Q2) For any x,y ∈ X, there is a unique z ∈ X such that z∗ y = x.(Q3) For any x,y,z ∈ X, (x∗ y)∗ z = (x∗ z)∗ (y∗ z).

In (Q2), the unique element z is denoted by x∗ y, and thenx = z∗ y = (x∗ y)∗ y.


Example(1) Let R3 = {0,1,2}. The binary operation ∗ : R3 ×R3 → R3 is as

the following table. Then R3 is a quandle.

∗ 0 1 20 0 2 11 2 1 02 1 0 2

(2) Let S4 = {0,1,2,3}. The binary operation ∗ : S4 ×S4 → S4 isas the following table. Then, S4 is a quandle.

∗ 0 1 2 30 0 2 3 11 3 1 0 22 1 3 2 03 2 0 1 3


Quandle coloring invariants of orientedsurface-linksLet B be a broken surface diagram of an oriented surface-linkL with co-orientation. A coloring of B is a functionC : S(B)→ X, where S(B) is the set of sheets of B and X is afinite quandle, satisfying the following conditions at double pointcurves.At a double point curve, let the co-orientation of over-sheet x isfrom y to z. Then C (z) = C (y)∗C (x). Let ColX(B) be the set ofall colorings of a broken surface diagram B by X.

yx

z


Theorem (Rosicki)Let L be an oriented surface-link and let B be a broken surfacediagram of L . Then the cardinality, #ColX(B), of ColX(B) is aninvariant of L , which is called a quandle coloring invariant of Land denoted by #ColX(L ).


Quandle coloring invariants #ColX(L ) viamarked graph diagramsLet Γ be a marked graph diagram with co-orientation. Let A (Γ)be the set of the connected components of Γ. A coloring of Γ isa function C : A (Γ)→ X satisfying the following condition:For each crossing c ∈ C(Γ), let s2 contain the over arc and let s1and s3 contain the under arcs involved in the crossing c asshown below such that the normal of the over arc in s2 pointsfrom the arc in s1 to the arc in s3. Then it must be satisfied thatC (s3) = C (s1)∗C (s2).

⌞

⌟

⌟s3

s1

s2 c⌞

⌟⌜

s3

s1

s2 c

We denote by ColX(Γ) the set of all X-colorings of Γ.Jieon Kim (Jointly with S. Kamada and S. Y. Lee) (PNU)Calculation of quandle cocycle invariants via marked graph diagramsAugust 25, 2014 12 / 25

Theorem (Ashihara)Let L be an oriented surface-link and Γ a marked graphdiagram of L . Then

#ColX(L ) = #ColX(Γ).


Let L be a surface-link. The ch-index χ(L ) of L is defined byminΓ χ(Γ), where Γ is a marked graph diagram of L andχ(Γ) = #C(Γ)+#V(Γ).

ExampleLet L be an oriented surface-link with χ(L )≤ 10.

L #ColR3(L ) #ColS4(L ) L #ColR3(L ) #ColS4(L )

01 3 4 102 9 421

1 3 4 103 3 1660,1

1 3 4 1011 9 16

81 9 16 100,11 3 4

81,11 3 4 100,1

2 3 491 9 16 101,1

1 3 490,1

1 3 4 100,0,11 9 16

101 3 4


>>

>

>>

>

> >

>

>

>

>

>

>

>

>

>

>

>>

>>

>

>

>

>

>>

01 21

>>

>>

>

>

>

>

>

>

>

>

>

>>

>

>

>>>

>

>

> >

>>

>

>

>

>

>

>

> >

>

>

>>

> >>>

> >

>

>

>

>

> >

>

>>

>

>

>

>>

>

> >

>

>

>

<

>

>

>

<

>

1 610,1 81 81

1,1

91 910,1 101 102

1011 101

0,1103

1020,1 101

1,1 1010,0,1


Contents

1 Surface-links




Quandle 3-cocyclesLet X be a finite quandle and A an abelian group with the identityelement 1. Carter-Jelsovsky-Kamada-Langford-Saito definedquandle homology group HQ

∗ (X;A) and the quandle cohomologygroup H∗

Q(X;A).

Note that a quandle 3-cocycle f : X×X×X → A satisfies(1) f (x,x,y) = 1 and f (x,y,y) = 1 for all x,y ∈ X.(2) f (x,y,w)f (x∗ y,z,w)f (x∗w,y∗w,z∗w)

= f (x,z,w)f (x,y,z)f (x∗ z,y∗ z,w), for each x,y,z,w ∈ X.

We fix a finite quandle X, an abelian group A and a 3-cocycleθ : X×X×X → A.


Quandle 3-cocycle invariants of orientedsurface-linksLet B be an oriented broken surface diagram of a surface-linkL and a coloring C of B be given.

c

b*C

aa*b

b

(a*b)*c

a*c

R c

b*C

aa*b

b

(a*b)*c

a*cR


The partition function of B is defined by

Φθ (B) = ∑C

∏τ

B(τ,C ) ∈ Z[A],

where the product is taken over all triple points, and the sum istaken over all possible colorings.

Theorem (Carter-Jelsovsky-Kamada-Langford-Saito)Let L be an oriented surface-link and let B be a broken surfacediagram of L . Then the partition function Φθ (B) is an invariantof L , which is called a quandle 3-cocycle invariant of L anddenoted by Φθ (L ).


Quandle 3-cocycle invariants Φθ(L ) viamarked graph diagramsLet Γ be a marked graph diagram of an oriented surface-link Land Γ+ (resp, Γ−) a positive (resp, negative) resolution of amarked graph diagram Γ. Let Γ+ = D1 → ··· → Dr = O (resp,Γ− = D′

1 → ·· · → D′s = O′) be a sequence of link diagrams from

Γ+ to O (resp, from Γ− to O′), related by Reidemeister movesand plane isotopies, where O and O′ are trivial diagrams. Let

Γ3 = {i|Di → Di+1 is a Reidemeister move of type 3},

Γ3′ = {j′|D′

j → D′j+1 is a Reidemeister move of type 3}.


ab

τ τ τ τR’

R’R’ R’

R R RRa

b

c c

cτ τ τ τ

R’ R’ R’ R’

R RR

Raa

b b

c

εtm(x)=1εb(x)=1

εtm(x)=-1εb(x)=1

Di(or Dj’) Di+1(or Dj+1’) Di(or Dj’) Di+1(or Dj+1’)

εtm(x)=1εb(x)=-1

εtm(x)=-1εb(x)=-1

Di(or Dj’) Di+1(or Dj+1’) Di(or Dj’) Di+1(or Dj+1’)


Definition(1) Let C : A (Γ)→ X be a coloring of Γ. Let x ∈ Γ3 ∪Γ3′. The

weight B(x,C ), for x ∈ Γ3 ∪Γ3′, is defined byθ(a,b,c)εr(x)εtm(x)εb(x), where εr : Γ3 ∪Γ3′ →{1,−1} is thefunction defined by εr(x) = 1 if x ∈ Γ3 and εr(x) =−1otherwise.

(2) The partition function of a marked graph diagram is definedby the state-sum expression

Φθ (Γ) = ∑C

∏x∈Γ3∪Γ3′

B(x,C ).

TheoremLet L be an oriented surface-link and let Γ be a marked graphdiagram of L . Then Φθ (L ) = Φθ (Γ).


ExampleLet L be an oriented surface-link such that χ(L )≤ 10. Let

θ = χ0,1,0−1

χ0,2,0χ0,2,1−1

χ1,0,1χ1,0,2χ2,0,2χ2,1,2 ∈ Z3Q(R3;Z3),

where χx,y,z(a,b,c) = u if (x,y,z) = (a,b,c), χx,y,z(a,b,c) = 1otherwise, and Z3 =< u|u3 = 1 > is the cyclic group. Let

η =χ0,1,0χ0,2,1χ0,2,3χ0,3,0χ0,3,1χ0,3,2χ1,0,1

χ1,0,3χ1,2,0χ1,3,1χ2,0,3χ2,1,0χ2,1,3χ2,3,2 ∈ Z3Q(S4;Z2),

where χx,y,z(a,b,c) = t if (x,y,z) = (a,b,c), χx,y,z(a,b,c) = 1otherwise, and Z2 =< t|t2 = 1 > is the cyclic group. Then Φθ (L )and Φη(L ) are as follows:


Example (Continued)L Φθ (L ) Φη(L ) L Φθ (L ) Φη(L )

01 3 4 102 3+6u 421

1 3 4 103 3 4+12t60,1

1 3 4 1011 9 16

81 9 16 100,11 3 4

81,11 3 4 100,1

2 3 491 9 16 101,1

1 3 490,1

1 3 4 100,0,11 9 16

101 3 4


�T�h�a�n�k� �y�o�u


Documents

Calculation of quandle cocycle invariants via marked …gt.postech.ac.kr/satellite2014/static/slides/jieon_kim.pdf · Jieon Kim (Jointly with S. Kamada and S. Y. Lee) (PNU)Calculation