Calculation of quandle cocycle invariants via marked graph diagrams

Transcription

1 Calculation of quandle cocycle invariants via marked graph diagrams Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Pusan National University, Busan, Korea August 25, 204 Knots and Low Dimensional Manifolds Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, 204 / 25

2 Contents Surface-links 2 Quandle coloring invariants of oriented surface-links 3 Quandle cocycle invariants of oriented surface-links Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

3 Contents Surface-links 2 Quandle coloring invariants of oriented surface-links 3 Quandle cocycle invariants of oriented surface-links Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

4 Surface-links A surface-link is a closed surface smoothly embedded in R 4. If a surface-link is oriented, then we call it an oriented surface-link. A marked graph diagram Γ Γ+ Γ- Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

5 A marked graph diagram Γ is said to be admissible if both resolutions Γ + and Γ are link diagrams of trivial links. Theorem (Kearton-Kurlin, Swenton, Yoshikawa) () 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. Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

6 Γ 0 Γ+ Γ Γ- - Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

7 Contents Surface-links 2 Quandle coloring invariants of oriented surface-links 3 Quandle cocycle invariants of oriented surface-links Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

8 Quandles A quandle is a set X with a binary operation : X X X satisfying (Q) 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 ȳ, and then x = z y = (x ȳ) y. Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

9 Example () Let R 3 = {0,,2}. The binary operation : R 3 R 3 R 3 is as the following table. Then R 3 is a quandle (2) Let S 4 = {0,,2,3}. The binary operation : S 4 S 4 S 4 is as the following table. Then, S 4 is a quandle Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

10 Quandle coloring invariants of oriented surface-links Let B be a broken surface diagram of an oriented surface-link L with co-orientation. A coloring of B is a function C : S(B) X, where S(B) is the set of sheets of B and X is a finite quandle, satisfying the following conditions at double point curves. At a double point curve, let the co-orientation of over-sheet x is from y to z. Then C (z) = C (y) C (x). Let Col X (B) be the set of all colorings of a broken surface diagram B by X. y x z Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

11 Theorem (Rosicki) Let L be an oriented surface-link and let B be a broken surface diagram of L. Then the cardinality, #Col X (B), of Col X (B) is an invariant of L, which is called a quandle coloring invariant of L and denoted by #Col X (L ). Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, 204 / 25

12 Quandle coloring invariants #Col X (L ) via marked graph diagrams Let 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 is a function C : A (Γ) X satisfying the following condition: For each crossing c C(Γ), let s 2 contain the over arc and let s and s 3 contain the under arcs involved in the crossing c as shown below such that the normal of the over arc in s 2 points from the arc in s to the arc in s 3. Then it must be satisfied that C (s 3 ) = C (s ) C (s 2 ). s 2 s c s 3 We denote by Col X (Γ) the set of all X-colorings of Γ. Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25 s 2 s c s 3

13 Theorem (Ashihara) Let L be an oriented surface-link and Γ a marked graph diagram of L. Then #Col X (L ) = #Col X (Γ). Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

14 Let L be a surface-link. The ch-index χ(l ) of L is defined by min Γ χ(γ), where Γ is a marked graph diagram of L and χ(γ) = #C(Γ) + #V(Γ). Example Let L be an oriented surface-link with χ(l ) 0. L #Col R3 (L ) #Col S4 (L ) L #Col R3 (L ) #Col S4 (L ) , , 3 4 8, , , , ,0, Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

15 0 2 < < 6 0, 8 8, 9 9 0, , , 0, 0 0,0, Jieon Kim (Jointly with S. Kamada and S. Y. Lee) (PNU) Calculation of quandle cocycle invariants via marked graph diagrams August 25, / 25

16 Contents Surface-links 2 Quandle coloring invariants of oriented surface-links 3 Quandle cocycle invariants of oriented surface-links Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

17 Quandle 3-cocycles Let X be a finite quandle and A an abelian group with the identity element. Carter-Jelsovsky-Kamada-Langford-Saito defined quandle homology group H Q (X;A) and the quandle cohomology group H Q (X;A). Note that a quandle 3-cocycle f : X X X A satisfies () f (x,x,y) = and f (x,y,y) = 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-cocycle θ : X X X A. Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

18 Quandle 3-cocycle invariants of oriented surface-links Let B be an oriented broken surface diagram of a surface-link L and a coloring C of B be given. (a*b)*c b*c b*c a*c (a*b)*c a*c a*b a a*b a b c b c Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

19 The partition function of B is defined by Φ θ (B) = C τ B(τ,C ) Z[A], where the product is taken over all triple points, and the sum is taken over all possible colorings. Theorem (Carter-Jelsovsky-Kamada-Langford-Saito) Let L be an oriented surface-link and let B be a broken surface diagram of L. Then the partition function Φ θ (B) is an invariant of L, which is called a quandle 3-cocycle invariant of L and denoted by Φ θ (L ). Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

20 Quandle 3-cocycle invariants Φ θ (L ) via marked graph diagrams Let Γ be a marked graph diagram of an oriented surface-link L and Γ + (resp, Γ ) a positive (resp, negative) resolution of a marked graph diagram Γ. Let Γ + = D D r = O (resp, Γ = D D s = O ) be a sequence of link diagrams from Γ + to O (resp, from Γ to O ), related by Reidemeister moves and plane isotopies, where O and O are trivial diagrams. Let Γ 3 = {i D i D i+ is a Reidemeister move of type 3}, Γ 3 = {j D j D j+ is a Reidemeister move of type 3}. Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

21 a R R R a R b b τ c τ c τ τ R R R R Di(or Dj ) Di+(or Dj+ ) Di(or Dj ) Di+(or Dj+ ) εtm(x)= εb(x)= εtm(x)=- εb(x)= a R b R a R R R b τ c τ c τ τ R R R Di(or Dj ) Di+(or Dj+ ) Di(or Dj ) Di+(or Dj+ ) εtm(x)= εb(x)=- εtm(x)=- εb(x)=- Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

22 Definition () Let C : A (Γ) X be a coloring of Γ. Let x Γ 3 Γ 3. The weight B(x,C ), for x Γ 3 Γ 3, is defined by θ(a,b,c) ε r(x)ε tm (x)ε b (x), where ε r : Γ 3 Γ 3 {, } is the function defined by ε r (x) = if x Γ 3 and ε r (x) = otherwise. (2) The partition function of a marked graph diagram is defined by the state-sum expression Φ θ (Γ) = C x Γ 3 Γ 3 B(x,C ). Theorem Let L be an oriented surface-link and let Γ be a marked graph diagram of L. Then Φ θ (L ) = Φ θ (Γ). Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

23 Example Let L be an oriented surface-link such that χ(l ) 0. Let θ = χ 0,,0 χ 0,2,0 χ 0,2, χ,0, χ,0,2 χ 2,0,2 χ 2,,2 Z 3 Q (R 3;Z 3 ), where χ x,y,z (a,b,c) = u if (x,y,z) = (a,b,c), χ x,y,z (a,b,c) = otherwise, and Z 3 =< u u 3 = is the cyclic group. Let η =χ 0,,0 χ 0,2, χ 0,2,3 χ 0,3,0 χ 0,3, χ 0,3,2 χ,0, χ,0,3 χ,2,0 χ,3, χ 2,0,3 χ 2,,0 χ 2,,3 χ 2,3,2 Z 3 Q (S 4;Z 2 ), where χ x,y,z (a,b,c) = t if (x,y,z) = (a,b,c), χ x,y,z (a,b,c) = otherwise, and Z 2 =< t t 2 = is the cyclic group. Then Φ θ (L ) and Φ η (L ) are as follows: Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

24 Example (Continued) L Φ θ (L ) Φ η (L ) L Φ θ (L ) Φ η (L ) u t 6 0, , 3 4 8, , , , ,0, Jieon Kim (Jointly with S. Kamada and S. Y. Lee) Calculation (PNU) of quandle cocycle invariants via marked graph diagrams August 25, / 25

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