Quantum Uq

December 20, 2019

Quantum U

(g) invariants of virtual knots

Sukuse Abe

Osaka City University Advanced Mathematical Institute

Background

Problem of invariants of virtual knots

• We know few non-trivial Finite type invariants.

• The relation of polynomial invariants is not clear.

• Quantum invariants were not defined.

⇒ We define new quantum invariants.

Today’s talk

• Definition of virtual knots

• Definition of quantum U_q(g) invariants

• Outline of proof

• Remark

• Example

• Universal quantum invariants

Definition of virtual knots

Figure1 Gauss diagrams

[ Goussarov, M., Polyak, M., Viro, O., (2000)]

=

=

=

=

= =

Figure2 The Reidemeister moves among Gauss diagrams

Definition of quantum U

( g ) invariants

A(S¹):a chord diagram. A(S¹)/ ∼:a target set

= 0, = = 0,

The AS relation, The IHX relation and The STU relation.

G := span_C{Gauss diagrams on S¹}, J : G → G

Figure3 The map J

=

Figure4 Example of the map J

I : G → A(S¹)/ ∼

Figure5 The map I

Remark 1. New terms in I ◦ J(g), the image of g ∈ G, never vanish.

Theorem 2. g^，g^′ ∈ G 

g ∼ g^′ ⇒ I ◦ J(g) = I ◦ J(g^′)

Outline of proof

I ◦ J

( )

= ε + = .

I ◦ J

( )

= − + ε − ε +

Lemma 3.

= ∈ A(S¹)/ ∼

Since

= ,

We obtain

=

=

Corollary 4. Let W_g,R : A(S¹)/ ∼→ C/ ∼^′ [ℏ] be a graded weight system. Then,

W_g,R ◦ I ◦ J(g) is an invariant of virtual knot g ∈ G.

Remark

Theorem 5. W_sl2,Vn : A(S¹)/ ∼→ C[ℏ]/ ∼^′is trivial.

Example

W_{soN ,C}^N

( )

= N − 1

2 W_{soN ,C}^N

( ) .

W_soN,C^N

( )

= W_{soN ,C}^N

( )

+ (N − 2)W_soN,C^N

( ) .

Remark 6. The map

P : G −→^J G −→ A^I (S¹)

W_so

N ,CN

−−−−−−→ C[N, ℏ] −−−→^N→2 C[ℏ] is not an invariant of virtual knot g ∈ G.

We consider following an abelian group

(C/ ∼, ×): a, b ∈ C, a ∼ b ⇔ ∃n ∈ Z s.t.

a/b = 2ⁿ, then (C/ ∼, ×) is well-defined.

Example 7. 1 ∼ 1/2 ∼ 1/4 ∼ 1/8 . . . ∈ C/ ∼. We put following relation

W_soN,C^N

( )

= 1.

Theorem 8. The map

Q : G −→^P C[ℏ] quotient mapping

−−−−−−−−−−−−−→ C/ ∼ [ℏ] is an invariant of virtual knot g ∈ G.

Example 9.

P

( )

= − 3

8 ℏ⁴ − 1

4 ℏ³ + 1ℏ + 1 ∈ C[ℏ], Q

( )

= − 3

8 ℏ⁴ ≁ −1ℏ⁴

∈ C/ ∼ [ℏ]/(The polynomials with coefficient 1).

⇒

̸

=

Universal quantum invariants

Corollary 10. The set {W_g,R ◦ I ◦ J(g) | g, R} is the universal quantum invariants of the virtual knot g ∈ G.

Conjecture 11. Universality of the

{W_g,R ◦ I ◦ J(g) | g, R} among various polynomial invariants of virtual knots.

Thank you!