Knot theory in 3-manifold via virtual knot theory

Teruhisa Kadokami (Kanazawa University)

Mathematics of Knots II December 20th, 2019 (Fri.)

Nihon University, College of Humanities and Sciences

Contents

§ 0. Introduction 4 – 4

Part I : Geometric virtual knot theory

§ 1. Diagrammatic definition of virtual link 6 – 13

§ 2. Geometric translation : Kuperberg’s theorem 14 – 22

Part II : Knot theory in 3-manifold

§ 3. Compression body decomposition of compact 3-manifolds 24 – 27

§ 4. Knot theory in 3-manifold via virtual knot theory 28 – 30

§ 5. Problems 31 – 31

References

[1] L. Kauffman Virtual knot theory

European Jounal of Combinatorics 20 (1999), 663–690.

[2] N. Kamada and S. Kamada

Abstract link diagrams and virtual knots

J. Knot Theory Ramifications 9 (2000), 93–106.

[3] S. Carter, S. Kamada and M. Saito

Stable equivalence of knots on surfaces and virtual knot cobordisms J. Knot Theory Ramifications 11 (2002), 311–322.

[4] G. Kuperberg

What is a virtual link ?

Algebraic Geometry & Topology 3 (2003), 587–591.

[5] T. Kadokami

Classification of closed virtual 2-braids

Journal of Knot Theory and its Ramifications 17 (2008), 1223–1239.

§ 0. Introduction

• Classical knot theory

link in S ³ = ⇒ diagram on S ² = ⇒ invariant Virtual knot theory

? = ⇒ diagram on S ² = ⇒ invariant

diagram on S ² = ⇒ diagram on F = ⇒ link in F × [0, 1] = ?

• M : ori. conn. compact 3-manifold

M = V ∪ W : Heegaard splitting F = V ∩ W : Heegaard surface

L : link in M = ⇒ L : link in N (F ) ∼ = F × [0, 1] ⊂ M

Part I : Geometric virtual knot theory

§ 1. Diagrammatic definition of virtual link Link in S ³ (Classical link)

• φ :

⨿ n

i=1

(S ¹ ) _i → S ³ or R ³ : embedding

= ⇒ L = Im(φ) = K ₁ ∪ . . . ∪ K _n : n-component link K _i = φ((S ¹ ) _i ) : the i-th component of L

◦ n = 1 = ⇒ L = K : knot

◦ ^∀ (S ¹ ) _i : oriented = ⇒ L : oriented link

• L, L ^′ : two links are equivalent (ambient-isotopic) ⇐⇒

∃ F : S ³ × [0, 1] → S ³ × [0, 1] : level-preserving homeo. s.t.

F ₀ = id _S

& F ₁ (L) = L ^′ . (i.e. F : ambient-isotopy)

Projection

p : S ³ = R ³ ∪ {∞} → S ² = R ² ∪ {∞} : projection (1) p((x, y, z)) = (x, y) if (x, y, z) ∈ R ³

(2) p( ∞ ) = ∞

c c

c

α

β

p

p (α) p (β) z

O

Link diagram

trefoil   3 figure eight knot   4 Hopf link   ^H

1 2 n

trivial knot O

trivial link O ⁿ

1

1

Reidemeister moves

(R1)

(R2)

(R3)

Theorem (Fundamental Theorem of Knot Theory) L, L ^′ : two links

D, D ^′ : two diagrams of L, L ^′ , respectively L ∼ = L ^′ : equivalent ⇐⇒

D ←→ D ^′ : finite sequence of Reidemeister moves L = { links } ⊃ L ⁿ = { n-component links }

D = { link diagrams } ⊃ D ⁿ = { n-component link diagrams }

L = D / ⟨ (R1), (R2), (R3) ⟩ ⊃ L ⁿ = D ⁿ / ⟨ (R1), (R2), (R3) ⟩

Φ : D → L , Φ _n : D ⁿ → L ⁿ : natural projections

Virtual link diagram

real crossing virtual crossing

virtual trefoil virtual Hopf link Kishino’s knot

Virtual Reidemeister moves

(R1)

(R2)

(R3)

(V1)

(V2)

(V3)

(V4)

V = { virtual links } ⊃ V ⁿ = { n-component virtual links } D = { virtual link diagrams }

⊃ D ⁿ = { n-component virtual link diagrams }

V = D / ⟨ (R1), (R2), (R3), (V1), (V2), (V3), (V4) ⟩

⊃ V ⁿ = D ⁿ / ⟨ (R1), (R2), (R3), (V1), (V2), (V3), (V4) ⟩

§ 2. Geometric translation : Kuperberg’s theorem Abstract link diagram [N. Kamada-S. Kamada]

D ( N ( D ~ ), D ~ )

D : virtual link diagram

(N ( D), e D) e : the abstract link diagram of D,

where N ( D) e : ori. compact surface canonically obtained from D, and

D e : a diagram on N ( D). e

Surface realization

D ( F , D ~ )

D : virtual link diagram

(N ( D), e D) e : the abstract link diagram of D (F, D) e : a surface realization of D,

where F : ori. closed surface obtained from N ( D) e by attaching compact surfaces to ∂N ( D e ).

(F, D) e : the canonical realization of D,

where F : ori. closed surface obtained from N ( D) e

by attaching disks to ∂N ( D e ).

Numerical invariants L : virtual link

D : diagram of L

(F, D) e : the canonical realization of D

sg(D) = (the sum of genera of components of F ) : the supporting genus of D

c(D) = (the number of components of F ) : the splitting number of D

sg(L) = min { sg(D) | D : diagram of L } : the supporting genus of L

c(L) = max { c(D ) | D : diagram of L } : the splitting number of L

Minimal realization L : virtual link

D : diagram of L

(F, D) e : a surface realization of D : minimal realization of L ⇐⇒

g (F ) = sg(L) & (the number of components of F ) = c(L).

Then D : minimal diagram.

Space realization L : virtual link D : diagram of L

(F, D) e : a surface realization of D

(F, D) e can be regarded as a (framed) link D b in F × [0, 1].

(F × [0, 1], D b ) or (F × [0, 1], L) b : space realization of (F, D). e If (F, D) e : the canonical realization of D = ⇒

(F × [0, 1], D b ) : space realization of D.

If (F, D) e : minimal realization of D = ⇒

(F × [0, 1], D b ) : minimal (space) realization of L.

Theorem [Kuperberg]

L : virtual link (1) (existence)

∃ D : minimal diagram of L (2) (uniqueness)

D, D ^′ : two minimal diagrams of L

(F, D), e (F ^′ , D f ^′ ) : the canonical realizations of D and D ^′ , respectively

= ⇒ (F × [0, 1], D b ), (F ^′ × [0, 1], D c ^′ ) : equivalent links

( ⇐⇒ (F, D), e (F ^′ , D f ^′ ) are related by an ori.-pres. homeo. φ : F → F ^′

& Reidemeister moves on F ^′

(i.e. φ( D e ) and D f ^′ are Reidemeister equivalent on F ^′ ).)

(H) : attaching a hollow handle to F \ D. e (H)

Theorem We can obtain a minimal realization by a finite sequence of (H) ^{− 1} -moves, and Reidemeister moves on the surface.

Remark L : virtual link, (F × [0, 1], L) b : space realization of L φ : F × [0, 1] → ( − F ) × [1, 0] : natural ori.-pres. homeo.

L ^♯ : virtual link determined from (F × [0, 1], φ( L)) b

: mixed mirror image of L. Then, in general, L ̸∼ = L ^♯ .

To regard (F × [0, 1], L) b as a virtual link, we should fix an ori. of F .

Part II : Knot theory in 3-manifold

§ 3. Compression body decomposition of compact 3-manifolds V : compression body

⇐⇒ V : tubing (surface) × [0, 1]’s and/or 3-balls by 1-handles

⇐⇒ V = (handle body) \ (standard sub-handle bodies) : dual def.

M : connected compact oriented 3-manifold

M = V ∪ W : ^∃ compression body decomposition of M F = V ∩ W : Heegaard surface

F = ∂ ₊ V = ∂ ₊ W

∂ ₋ V = ∂V \ F, ∂ ₋ W = ∂W \ F

M = V ∪ W : ordered compression body decomposition of M

Moves of compression body decompositions (C1) : (stabilization)

(C2) : (tubing along a trivial arc) M = V ∪ W −→

Σ : a component of ∂ ₋ W

V ^′ : tubing V and N (Σ) along a trivial arc α in W W ^′ = M \ V ^′

V ∪ W ←→ V ^′ ∪ W ^′

(C3) : (interchanging) V ∪ W ←→ W ∪ V

Theorem Compression body decompositions of M are related by a

finite sequence of (C1), (C2) and (C3).

(C1)

V V

W W

(C 2 )

V W

V α W

F F

F F

H = { ordered compression body decomp.s }

M = { connected compact oriented 3-manifolds }

= H / ⟨ (C1), (C2), (C3) ⟩ M f = H / ⟨ (C1), (C2) ⟩

H −→ ^p M f −→ M ^p

: natural projections

H 0 = { minimal genus ordered compression body decomp.s } ⊂ H

∀ M ∈ M , we take sets

D (M ) = (p ^′ ◦ p) ^{− 1} (M ) ⊃ D 0 (M ) = D (M ) ∩ H 0 ̸ = ∅ .

ex. M = F × [0, 1] = ⇒ D ⁰ (M ) = { 2 points } .

§ 4. Knot theory in 3-manifold via virtual knot theory M : connected compact oriented 3-manifold

L = K ₁ ∪ . . . ∪ K _n ⊂ M : link in M

M = V ∪ W : ordered compression body decomposition of M F = V ∩ W : Heegaard surface

L can be regarded as a link in N (F ) ≒ virtual link M = N (F ) ∪ (2-handles and 3-handles).

“Main Theorem”

{ links in M } ←→ { links in N (F ) } / ⟨ (C1), (C2), (C3), 2-handles ⟩

L ⊂ M −→ L ⊂ N (F )

−→ L as a virtual link : a representing virtual link For L : link in M , s = [M = V ∪ W ] ∈ D (M ), V (L, s) : the representing virtual links of L in s.

V (L, M ) = ∪

s ∈D (M )

V (L, s), V 0 (L, M ) = ∪

s ∈D

(M )

V (L, s),

V (M ) = ∪

L

V (L, M ), V ⁰ (M ) = ∪

L

V (L, M ).

s = [V ∪ W ] −→

s ^♯ = [W ∪ V ].

Lemma K ∈ V (L, s) = ⇒ K ^♯ ∈ V (L, s ^♯ ).

Lemma V (S ³ ) = V (D ³ ) = { virtual links } (= V ).

V b (L, s) = { ℓ ∈ V (L, s) | sg(ℓ) = min { sg(ℓ ^′ ) | ℓ ^′ ∈ V (L, s) }}

V b (L, M ) = ∪

s ∈D (M )

V b (L, s), V b 0 (L, M ) = ∪

s ∈D

(M )

V b (L, s),

V b (M ) = ∪

L

V (L, M ), V b ⁰ (M ) = ∪

L

V (L, M ).

Lemma s = [M = V ∪ W ] ∈ D (M ) & V or W : handlebody

= ⇒ V b (L, s) consists of classical links.

ex. (1) M : lens space

T : torus knot in M = ⇒ V b ⁰ (T, M ) = { O } .

(2) V b ⁰ (L, F × [0, 1]) = { 1 point } .

§ 5. Problems

Q1. Determine V (L, M ), V (M ), V ⁰ (L, M ), V ⁰ (M ), V b (L, M ), V b (M ), V b ⁰ (L, M ), V b ⁰ (M ).

Q2. Is V b 0 (L, M ) finite in general ?

In particular, ♯ V b ⁰ (L, M ) = 1 in general ?

Q3. Does V b (L, M )/ V b 0 (L, M ) characterize (M, L) ? Q4. Does V b (M )/ V b 0 (M ) characterize M ?

Q5. Invariant study.

Q6. fundamental group −→ Gordon-Lueke type theorem ?

Thank you for your attention !