Biquandle cocycle invariants from marked graphs Seiichi Kamada (Osaka City University)

Biquandle cocycle invariants from marked graphs

Seiichi Kamada (Osaka City University) ^{∗ 1}

Akio Kawauchi (Osaka City University Advanced Mathematical Institute ) ^{∗ 2} Jieon Kim (Osaka City University) ^∗3

Sang Youl Lee (Pusan National University) ^{∗ 4}

1. Representations of 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 broken surface diagram of a surface-link L in R ⁴ is a generic surface of L into R ³ with over/under sheet information at each double curve.

(a) (b) (c) (d)

A marked graph is a finite spatial regular graph with 4-valent rigid vertices such that each vertex has a marker. A diagram of a marked graph in R ² is called a marked graph diagram or ch-diagram.

Γ

L

(Γ)

>

>

>

>

>

>

>

>

L

(Γ)

A marked graph diagram is said to be admissible if both resolutions L ₊ (Γ) and L ₋ (Γ) are diagrams of trivial links.

The first and second authors were supported by JSPS KAKENHI Grant Numbers 26287013 and 24244005. The third author was supported by JSPS overseas post doctoral fellow Grant Number 15F15319. The fourth author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2016R1A2B4016029).

2000 Mathematics Subject Classification: 57M25, 57M27.

Keywords: marked graph diagram, biquandle cocycle invariant.

∗1

e-mail: [email protected]

∗2

e-mail: [email protected]

∗3

e-mail: [email protected]

∗4

e-mail: [email protected]

An orientation of a marked graph G in R ³ is a choice of an orientation for each edge of G in such a way that every rigid vertex in G looks like ⌞

⌝

⌜ ⌟ or ⌝

⌞

⌟

⌜

. A marked graph in R ³ is said to be orientable if it admits an orientation. Otherwise, it is said to be nonorientable.

> > > >

> > > >

Theorem 1.1. [4, 7]

(1) For any surface-link L , 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

-2

2 ^>

>

> > Γ

1=t

t

-1=t

’

t

’

(Γ)

L

(Γ) L

(Γ)

Theorem 1.2. [5, 6] Two marked graph diagrams represent the same surface-link if and only if they are transformed into each other by a finite sequence of Yoshikawa moves.

Ω

: Ω

: Ω

: Ω

:

Ω

: Ω

:

Ω

: Ω

: Ω

:

Ω

:

Definition 1.3. Let R ³ + = { (x 1 , x 2 , x 3 , x 4 ) | x 3 ≥ 0, x 4 = 0 } , and K a classical knot.

T K ⊂ R ³ is a tangle of K whose end points lie in R ² . Then { (x ₁ , x ₂ , x ₃ sinθ, x ₃ cosθ) | (x ₁ , x ₂ , x ₃ ) ∈ T k , θ ∈ [0, 2π) } forms a surface knot. We call it the spun K .

R ²

R ³ +

T K

Definition 1.4. While rotating R ³ + about R ² , twist B ³ n times (n ∈ Z ). Then we have another surface knot. We call it the n twist spun K.

R ²

R ³ +

T K

B ³

Theorem 1.5. [2] The following marked graph diagram is a marked graph diagram of the n twist spun trefoil.

n-times

1-twist

2. Biquandle Cocycle Invariants

2.1. Biquandle Colorings

Definition 2.1. A biquandle X is a set with two binary operations ▷, ▷ : X × X → X such that

(1) For any x ∈ X, x▷x = x▷x.

(2) Two binary operations ▷, ▷ are right invertible.

(3) The map H : X × X → X × X defined by (x, y) 7→ (y▷x, x▷y) is invertible.

(4) For any x, y, z ∈ X,

(x▷y)▷(z▷y) = (x▷z)▷(y▷z), (x▷y)▷(z▷y) = (x▷z)▷(y▷z), (x▷y)▷(z▷y) = (x▷z)▷(y▷z).

Definition 2.2. Let X be a biquandle. A (biquandle) coloring on an oriented link diagram is a function C : S → X, where S is the set of semi-arcs in the diagram, satisfying the condition depicted in the figure below.

b

b

b

b b

Definition 2.3. A (biquandle) coloring on an oriented broken surface diagram is a function C : S → X, where S is the set of semi-sheets, satisfying the following condition at the double point set.

a c

b a

b

b

b b

b

c

c c c

(b ) (c )

= ( c) ( c)

=

( c) ( c) ( b) (c )

=

(c ) (b ) (c ) ( b)

Definition 2.4. Let Γ be an oriented marked graph diagram and X a finite biquandle.

A coloring of Γ is C : S(Γ) → X, where S(Γ) is the set of semi-arcs in Γ, satisfying the following conditions:

(1) For each crossing c ∈ C(Γ),

C (s 3 ) = C (s 1 )▷ C (s 2 ), C (s 4 ) = C (s 2 )▷ C (s 1 ).

⌝

⌞

⌟

⌟ s 4

s 3

s 1

s 2 c

⌞

⌟

⌜ ⌞

s 3

s 2

s 1

s 4 c

(2) For each marked vertex v ∈ V (Γ),

C (s 1 ) = C (s 2 ) = C (s 3 ) = C (s 4 ).

⌞

⌝

⌜ ⌟ s 4

s ₁ s 3

s ₂ v

⌞ ⌝

⌟

⌜ s 4

s ₁ s 3

s ₂

v We denote by Col X (Γ) the set of colorings of Γ.

Example 2.5. Let

M = [ { m ¹ _i,j } ¹ ≤ i,j ≤ 4 |{ m ² _i,j } ¹ ≤ i,j ≤ 4 ] =









1 4 2 3 1 1 1 1 2 3 1 4 3 3 3 3 3 2 4 1 4 4 4 4 4 1 3 2 2 2 2 2







 ,

and X = { 1, 2, 3, 4 } the biquandle determined by i▷j = m ¹ _i,j and i▷j = m ² _i,j . Let Γ n be a marked graph diagram of the n twist spun trefoil.

1

2

3

4

Γ

Hence #Col X (Γ _{3k −2} ) = #Col X (Γ _{3k −1} ) = 4, #Col X (Γ _3k ) = 4 + (4 × 3) = 16 for k ≥ 1.

2.2. Biquandle cocycles

Let X be a finite biquandle and A an abelian group with the identity element 1.

Carter, Elhamdadi, and Saito defined the biquandle homology group H _∗ ^Q (X; A) and the biquandle cohomology group H _Q ^∗ (X; A).

Note that a biquandle 2-cocycle f : C ₂ ^Q (X) → A satisfies (1) f(x, x) = 1 for all x, y ∈ X.

(2) f(y, z)f (x, y )f (x▷y, z▷y) = f (x, z)f (y▷x, z▷x)f(x▷z, y▷z), for each x, y, z ∈ X.

Note that a biquandle 3-cocycle f : C ₃ ^Q (X) → A satisfies (1) f(x, x, y) = 1 and f (x, y, y) = 1 for all x, y ∈ X.

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

Let D be an oriented diagram of a link L and a coloring C of D given. Let θ ∈ Z _Q ² (X; A).

b

b

b

θ (a, b) θ (a, b) ^{− 1}

b b

The partition function of D is defined by Φ θ (D) = X

C

Y

c

B θ (c, C ) ∈ Z [A].

Theorem 2.6. [1] Let L be a link and D a diagram of L. Then the partition function Φ θ (D) is an invariant of L, which is called the biquandle cocycle invariant of L and denoted by Φ θ (L).

Let B be an oriented diagram of a surface-link L and a coloring C of B given. Let θ ∈ Z _Q ³ (X; A).

a c

b b b

c

c c c

(b ) (c )

= ( c) ( c)

=

( c) ( c) ( b) (c )

=

(c ) (b ) (c ) ( b)

θ(a,b,c)

The partition function of B is defined by Φ θ ( B ) = X

C

Y

τ

B θ (τ, C ) ∈ Z [A].

Theorem 2.7. Let L be a surface-link and B a diagram of L . Then the partition function Φ θ ( B ) is an invariant of L , which is called the biquandle cocycle invariant of L and denoted by Φ θ ( L ).

Define I ₊ ³ = { i | D i → D i+1 is a Reidemeister move 3 } and I ₋ ³ = { j | D ^′ _j → D _j+1 ^′ is a Reidemeister move 3 } .

Let i ∈ I ₊ ³ . (resp., j ∈ I ₋ ³ .) Exactly one of D i and D i+1 (resp., D ^′ _j and D ^′ _i+1 ) has the region from which all normal orientations point outward such that the number of intersecting semi-arcs is 3. Let the region call the source region of i (resp., j ).

Definition 2.8. Let L be an oriented surface-link and Γ a marked graph diagram of L . Let C : S(Γ) → X be a coloring of Γ and θ ∈ Z _Q ³ (X; A).

(1) Let i ∈ I ₊ ³ . The (Boltzman) weight B θ (i, C ), for i ∈ I ₊ ³ , is defined by B θ (i, C ) = θ(x 1 , x 2 , x 3 ) ^ϵ

^(i)ϵ

⁽ⁱ⁾ ,

where x ₁ , x ₂ and x ₃ are colors of the bottom, middle and top arcs, respectively,

those bound the source region of i.

(2) Let j ∈ I ₋ ³ . The (Boltzman) weight B θ (j, C ), for j ∈ I ₋ ³ , is defined by B θ (j, C ) = θ(x 1 , x 2 , x 3 ) ^{− ϵ}

^(j)ϵ

^(j) ,

where x ₁ , x ₂ and x ₃ are colors of the bottom, middle and top arcs, respectively, those bound the source region of j.

ε

( ) c

=1 ε

( ) =-1

ε

( )=1 ε

( )=-1 c

c

c

Definition 2.9. Let Γ be a marked graph diagram of an oriented surface-link L . The partition function or state-sum (associated with θ) of a marked graph diagram Γ is defined by the state-sum expression

Φ θ (Γ) = X

C∈ Col

(Γ)

Y

x ∈ I

∪ I

B θ (x, C ),

where B θ (x, C ) is a weight of x ∈ I ₊ ³ ∪ I ₋ ³ .

Theorem 2.10. [3] Let L be an oriented surface-link and Γ a marked graph diagram of L . Then for any θ ∈ Z _Q ³ (X; A), Φ θ ( L ) = Φ θ (Γ).

Example 2.11. Let X be the biquandle in Example 2.5 and θ = χ _(1,4,1) χ _(1,4,3) χ _(2,4,1) χ (2,4,3) χ (3,2,1) χ (3,2,3) χ (4,2,1) χ (4,2,3) a cocycle with the coefficient Z ² =< t | t ² = 1 >, where χ (a,b,c) (x, y, z) is defined to be t if (x, y, z) = (a, b, c) and 1 otherwise.

L

(Γ

)

→ → →

→ → →

→ → →

L

(Γ

)

→ →

Then the biquandle cocycle invariant is Φ θ (Γ) = X

C∈ Col

(Γ)

Y

x ∈ I

∪ I

B θ (x, C ) = 4 + 12t.

3. Shadow Biquandle Cocycle Invariants

For a biquandle X, let G X =< x ∈ X; x▷y = y ⁻¹ xy (x, y ∈ X) > . An X-set is a set Y equipped with a right action of G X . Let y▷g be the image of y ∈ Y by the action g ∈ G X .

Note that a shadow biquandle 3-cocycle f : C ₃ ^Q (X) Y → A satisfies (1) f(a, x, x, y) = 1 and f (a, x, y, y) = 1 for all a ∈ Y, x, y ∈ X.

(2) f(a, y, z, w)f (a, x, y, w)f (a▷y, x▷y, z▷y, w▷y) f (a▷w, x▷w, y▷w, z▷w) = f (a, x, z, w) f(a, x, y, z) f(a▷x, y▷x, z▷x, w▷x)f (a▷z, x▷z, y▷z, w▷z), for each a ∈ Y and

x, y, z, w ∈ X.

Definition 3.1. Let X be a biquandle and let D be a diagram of an oriented link L.

Let C : S → X be a coloring of D. Let R be the set of the complementary regions of D in R ² . A shadow (biquandle) coloring of D (extending a given coloring C ) is a map C ˜ : S ∪ R → X satisfying the conditions:

• The restriction of ˜ C to S is the given coloring C .

• If two adjacent regions f 1 and f 2 are separated by a semi-arc e and the co- orientation of e points from f 1 to f 2 , then ˜ C (f 1 )▷ C ˜ (e) = ˜ C (f 2 ).

a

b a

b

x x

= a

( b) ( b) ( x a ) ( )

x =

a ( x a ) ( )

( x b) ( b) b

x

x b x a

x a

b b

b b

a

Theorem 3.2. [3] Let L be an oriented link and D be an oriented diagram of L.

Let Col ^S _X (D) be the set of all shadow colorings. Then the cardinality of Col ^S _X (D),

#Col ^S _X (D), is an invariant of L and denoted by #Col ^S _X (L).

Definition 3.3. Let X be a biquandle and let B be a diagram of an oriented surface- link L . Let C : S → X be a coloring of B . Let R be the set of the complementary regions of B in R ³ . A shadow (biquandle) coloring of B (extending a given coloring C ) is a map ˜ C : S ∪ R → X satisfying the conditions:

• The restriction of ˜ C to S is the given coloring C .

• If two adjacent regions f 1 and f 2 are separated by a semi-sheet e and the co- orientation of e points from f 1 to f 2 , then ˜ C (f 1 )▷ C ˜ (e) = ˜ C (f 2 ).

a b

x

= a

( b) ( b) ( x a ) ( a ) x

a x x b b

b

Definition 3.4. Let X be a biquandle and let Γ be a marked graph diagram of an oriented surface-link L . Let C : S → X be a coloring of Γ. Let R be the set of the complementary regions of Γ in R ² . A shadow (biquandle) coloring of Γ (extending a given coloring C ) is a map ˜ C : S ∪ R → X satisfying the conditions:

• The restriction of ˜ C to S is the given coloring C .

• If two adjacent regions f 1 and f 2 are separated by a semi-arc e and the co- orientation of e points from f 1 to f 2 , then ˜ C (f 1 )▷ C ˜ (e) = ˜ C (f 2 ).

a

b a

b

x x

= a

( b) ( b) ( x a ) ( )

x =

a ( x a ) ( )

( x b) ( b) b

x

b a x

x

a

b b x

b b

a

Theorem 3.5. [3] Let L be an oriented link and D be an oriented diagram of L.

Let Col ^S _X (D) be the set of all shadow colorings. Then the cardinality of Col ^S _X (D),

#Col ^S _X (D), is an invariant of L and denoted by #Col ^S _X (L).

Let ˜ C be a shadow coloring of D. Let θ ∈ Z _Q ² (X; A) Y .

Ө (x,a,b) x Ө (x,a,b)

x a

b a

b b

b b

b

The shadow partition function of D is defined by Φ ^S _θ (D) = X

C ˜

Y

c

B _θ ^S (c, C ˜ ) ∈ Z [A].

Theorem 3.6. [3] Let L be a link and D a diagram of L. Then the shadow partition function Φ ^S _θ (D) is an invariant of L, which is called the shadow biquandle cocycle invariant of L and denoted by Φ ^S _θ (L).

Let ˜ C be a shadow coloring of B . Let θ ∈ Z _Q ³ (X; A) Y .

Ө (x,a,b,c)

a c

b b b

c

c c c

(b ) (c )

= ( c) ( c)

=

( c) ( c) ( b) (c )

=

(c ) (b ) (c ) ( b)

x

The shadow partition function of B is defined by Φ ^S _θ ( B ) = X

C ˜

Y

τ

B _θ ^S (τ, C ˜ ) ∈ Z [A].

Theorem 3.7. [3] Let L be a surface-link and B a diagram of L . Then the shadow partition function Φ ^S _θ ( B ) is an invariant of L , which is called the shadow biquandle cocycle invariant of L and denoted by Φ ^S _θ ( L ).

Let ˜ C be a shadow coloring of Γ. Let θ ∈ Z _Q ² (X; A) Y .

x a

c b

Ө (x,a,b,c)

D

D

b b

c

=(c ) (b ) (c ) ( b)

=( c) ( c) ( b) (c )

c c

c a c

c b (b ) (c )

= ( c) ( c)

=

(c ) (b ) (c ) ( b)

=

( c) ( c) ( b) (c )

The shadow partition function of Γ is defined by Φ ^S _θ (Γ) = X

C ˜

Y

c

B _θ ^S (c, C ˜ ) ∈ Z [A].

Theorem 3.8. [3] Let L be an oriented surface-link and Γ a marked graph diagram

of L . Then for any θ ∈ Z _Q ³ (X; A) Y , Φ ^S _θ ( L ) = Φ ^S _θ (Γ).

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[3] S. Kamada, A. Kawauchi, J. Kim, and S. Lee, Biquandle cohomology and state-sum invariants of knots and surface-links, in preparation.

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3, 497-522.