Knot theory in 3 -manifold via virtual knot theory Teruhisa Kadokami - PowerPoint PPT Presentation

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

2/32 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

3/32 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.

4/32 § 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 N ( F ) ∼ = F × [0 , 1] ⊂ M L : link in M =

5/32 Part I : Geometric virtual knot theory

6/32 § 1. Diagrammatic definition of virtual link Link in S 3 (Classical link) n ⨿ ( S 1 ) i → S 3 or R 3 : embedding • φ : i =1 ⇒ 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 3 & F 1 ( L ) = L ′ . (i.e. F : ambient-isotopy)

7/32 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 ( ∞ ) = ∞ z c 1 α c 2 β p p (α) O c p (β)

8/32 Link diagram trivial knot O trefoil 3 figure eight knot 4 Hopf link H 1 1 1 2 n trivial link O n

9/32 Reidemeister moves (R1) (R2) (R3)

10/32 Theorem (Fundamental Theorem of Knot Theory) L, L ′ : two links D, D ′ : two diagrams of L , L ′ , respectively = L ′ : equivalent ⇐ L ∼ ⇒ → D ′ : finite sequence of Reidemeister moves D ← 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

11/32 Virtual link diagram real crossing virtual crossing virtual trefoil virtual Hopf link Kishino’s knot

12/32 Virtual Reidemeister moves (R1) (V1) (R2) (V2) (R3) (V3) (V4)

13/32 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) ⟩

14/32 § 2. Geometric translation : Kuperberg’s theorem Abstract link diagram [N. Kamada-S. Kamada] ~ ~ ( N ( D ), D ) D

15/32 D : virtual link diagram ( N ( � D ) , � D ) : the abstract link diagram of D , where N ( � D ) : ori. compact surface canonically obtained from D , and D : a diagram on N ( � � D ) .

16/32 Surface realization ~ ( F , D ) D

17/32 D : virtual link diagram ( N ( � D ) , � D ) : the abstract link diagram of D ( F, � D ) : a surface realization of D , where F : ori. closed surface obtained from N ( � D ) by attaching compact surfaces to ∂N ( � D ) . ( F, � D ) : the canonical realization of D , where F : ori. closed surface obtained from N ( � D ) by attaching disks to ∂N ( � D ) .

18/32 Numerical invariants L : virtual link D : diagram of L ( F, � D ) : 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

19/32 Minimal realization L : virtual link D : diagram of L ( F, � D ) : 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.

20/32 Space realization L : virtual link D : diagram of L ( F, � D ) : a surface realization of D ( F, � D ) can be regarded as a (framed) link � D in F × [0 , 1] . ( F × [0 , 1] , � D ) or ( F × [0 , 1] , � L ) : space realization of ( F, � D ) . If ( F, � D ) : the canonical realization of D = ⇒ ( F × [0 , 1] , � D ) : space realization of D . If ( F, � ⇒ D ) : minimal realization of D = ( F × [0 , 1] , � D ) : minimal (space) realization of L .

21/32 Theorem [Kuperberg] L : virtual link (1) (existence) ∃ D : minimal diagram of L (2) (uniqueness) D, D ′ : two minimal diagrams of L ( F, � D ) , ( F ′ , � D ′ ) : the canonical realizations of D and D ′ , respectively D ) , ( F ′ × [0 , 1] , � ⇒ ( F × [0 , 1] , � D ′ ) : equivalent links = ⇒ ( F, � D ) , ( F ′ , � D ′ ) are related by an ori.-pres. homeo. φ : F → F ′ ( ⇐ & Reidemeister moves on F ′ D ) and � D ′ are Reidemeister equivalent on F ′ ).) (i.e. φ ( �

22/32 (H) : attaching a hollow handle to F \ � D . (H) Theorem We can obtain a minimal realization by a finite sequence of (H) − 1 -moves, and Reidemeister moves on the surface. ( F × [0 , 1] , � Remark L : virtual link, L ) : 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 )) Then, in general, L ̸∼ = L ♯ . : mixed mirror image of L . To regard ( F × [0 , 1] , � L ) as a virtual link, we should fix an ori. of F .

23/32 Part II : Knot theory in 3 -manifold

24/32 § 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

25/32 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).

26/32 W W (C1) F F V V α W W (C 2 ) F F V V

27/32 H = { ordered compression body decomp.s } M = { connected compact oriented 3 -manifolds } = H / ⟨ (C1) , (C2) , (C3) ⟩ � M = H / ⟨ (C1) , (C2) ⟩ p ′ p → � H − M − → M : 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 } .

28/32 § 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 ⟩

29/32 L ⊂ M − → L ⊂ N ( F ) − → L as a virtual link : a representing virtual link s = [ M = V ∪ W ] ∈ D ( M ) , For L : link in M , V ( L, s ) : the representing virtual links of L in s . ∪ ∪ V ( L, M ) = V ( L, s ) , V 0 ( L, M ) = V ( L, s ) , s ∈D ( M ) s ∈D 0 ( M ) ∪ ∪ V ( M ) = V ( L, M ) , V 0 ( M ) = V ( L, M ) . L L (C3) → s ♯ = [ W ∪ V ] . s = [ V ∪ W ] − ⇒ K ♯ ∈ V ( L, s ♯ ) . K ∈ V ( L, s ) = Lemma V ( S 3 ) = V ( D 3 ) = { virtual links } (= V ) . Lemma

30/32 V ( L, s ) = { ℓ ∈ V ( L, s ) | sg ( ℓ ) = min { sg ( ℓ ′ ) | ℓ ′ ∈ V ( L, s ) }} � ∪ ∪ � � � � V ( L, M ) = V ( L, s ) , V 0 ( L, M ) = V ( L, s ) , s ∈D ( M ) s ∈D 0 ( M ) ∪ ∪ � � V ( M ) = V ( L, M ) , V 0 ( M ) = V ( L, M ) . L L s = [ M = V ∪ W ] ∈ D ( M ) & V or W : handlebody Lemma ⇒ � = V ( L, s ) consists of classical links. ex. (1) M : lens space ⇒ � V 0 ( T, M ) = { O } . T : torus knot in M = (2) � V 0 ( L, F × [0 , 1]) = { 1 point } .