An application of braid ordering to knot theory

Transcription

1 January 11, 2009

2 Plan of the talk I The Dehornoy ordering, Dehornoy floor I Dehornoy floor and genus of closed braid I Nielsen-Thurston classification and Geometry of closed braid

3 Braid groups I B n : Braid group of n-strands σ B n = σ 1,, σ i σ j σ i = σ j σ i σ j i j = 1 n 1 σ i σ j = σ j σ i i j 2 1 i i+1 n There is a natural isomorphism B n = MCG(Dn, D n ). ( D n = D 2 /{p-punctures} )

4 The Dehornoy ordering: Algebraic definition Definition: Dehornoy ordering For braids α, β B n, α < D β The braid α 1 β has word presentation such that { contains no σ ±1 1,, σ±1 contains σ i at least one i 1, σ 1 i

5 The Dehornoy ordering: Algebraic definition Definition: Dehornoy ordering For braids α, β B n, α < D β The braid α 1 β has word presentation such that { contains no σ ±1 1,, σ±1 i 1, σ 1 i contains σ i at least one The Dehornoy ordering is left-invariant total ordering. α < D β γα < D γβ for all γ B n

6 Examples I σ 2 < D σ 1 :

7 Examples I σ 2 < D σ 1 : The braid word (σ 1 2 )(σ 1) contains σ 1 but no σ 1 1.

8 Examples I σ 2 < D σ 1 : The braid word (σ 1 2 )(σ 1) contains σ 1 but no σ 1 1. I σ 1 < D σ 2 σ 1 :

9 Examples I σ 2 < D σ 1 : The braid word (σ2 1 )(σ 1) contains σ 1 but no σ1 1. I σ 1 < D σ 2 σ 1 : The braid (σ1 1 )σ 2σ 1 contains both σ 1 and σ1 1, but equivalent word σ 2 σ 1 σ2 1 contains σ 1 but no σ1 1.

10 Examples I σ 2 < D σ 1 : The braid word (σ2 1 )(σ 1) contains σ 1 but no σ1 1. I σ 1 < D σ 2 σ 1 : The braid (σ1 1 )σ 2σ 1 contains both σ 1 and σ1 1, but equivalent word σ 2 σ 1 σ2 1 contains σ 1 but no σ1 1. I σ1 1 σ 2 < D 1 but (σ1 1 σ 2)(σ 1 σ 2 σ 1 ) > D σ 1 σ 2 σ 1.

11 Examples I σ 2 < D σ 1 : The braid word (σ2 1 )(σ 1) contains σ 1 but no σ1 1. I σ 1 < D σ 2 σ 1 : The braid (σ1 1 )σ 2σ 1 contains both σ 1 and σ1 1, but equivalent word σ 2 σ 1 σ2 1 contains σ 1 but no σ1 1. I σ1 1 σ 2 < D 1 but (σ1 1 σ 2)(σ 1 σ 2 σ 1 ) > D σ 1 σ 2 σ 1. This means the Dehornoy ordering is not right invariant ordering.

12 The Dehornoy ordering:geometric definition Γ: Horizontal diameter of D n Identify B n = MCG(Dn, D n ) α < D β β(γ) moves more left than α(γ).

13 The Dehornoy ordering:geometric definition Γ: Horizontal diameter of D n Identify B n = MCG(Dn, D n ) α < D β β(γ) moves more left than α(γ). ½ µ ¾ µ ½ ¾

14 The Dehornoy floor Definition: Dehornoy floor β B n : n-braid The Dehornoy floor [β] D [β] D = min{i Z 0 2i 2 < β < 2i+2 } where = (σ 1 σ 2 σ n 1 )(σ 1 σ n 2 ) (σ 1 σ 2 )(σ 1 ): Garside fundamental braid ¾ ½ ¾ ½ ¼ ½ Ò The Dehornoy floor can be seen as a measure of complexity of braids.

15 Properties of Dehornoy floor Dehornoy floor has following property. Proposition I If β is conjugate to a braid which contains σ ±1 1 at most k, then [β] D < k I [αβ] D [α] D + [β] D + 1 I α,β conjugate [α] D [β] D 1. These properties are used to estimate Dehornoy floor via word representatives.

16 Simple Application of Dehornoy floor A B Exchange move A B Every closed braid representatives of unlinks or composite links admit exchange move. (Birman-Menasco) [β] D 0 implies closure of β is non-trivial and prime. (Malyutin)

17 Genus formula Previous theorem is generalized as follows. Theorem [Ito] β B n : n-braid K = β: oriented knot g(k): genus of K [β] D < 4g(K) n n

18 Genus formula Previous theorem is generalized as follows. Theorem [Ito] β B n : n-braid K = β: oriented knot g(k): genus of K [β] D < 4g(K) n n A closure of complex braid is complex.

19 The Nielsen-Thurston classification Braids β B n = MCG(Dn, D n ) are classified into following three types by their dynamics (Nielsen-Thurston): I Periodic: β N = 2p for some N. I Reducible: β preserves some essential curve on D n. I Pseudo-Anosov: β is represented by pseudo-anosov homeomorphisms.

20 Nielsen-Thurston classification and geometry F : Oriented closed surface, genus 2 ϕ : F F : Orientation preserving homeomorphism M ϕ = F I / ( (x, 0) (ϕ(x), 1)): Mapping torus of ϕ. Theorem (Thurston) I ϕ is periodic M ϕ is Seifert-fibered. I ϕ is reducible M ϕ has essential tori. I ϕ is pseudo-anosov M ϕ is hyperbolic.

21 Nielsen-Thurston classification and geometry F : Oriented closed surface, genus 2 ϕ : F F : Orientation preserving homeomorphism M ϕ = F I / ( (x, 0) (ϕ(x), 1)): Mapping torus of ϕ. Theorem (Thurston) I ϕ is periodic M ϕ is Seifert-fibered. I ϕ is reducible M ϕ has essential tori. I ϕ is pseudo-anosov M ϕ is hyperbolic. The Nielsen-Thurston classification and geometry of mapping torus are in one-to-one correspondence.

22 Geometry of knot complements Knots K are classified into three types via geometry of their complements E K = S 3 \K. I Torus knots: E K are Seifert-fibered. I Satellite knots: E K have essential tori. I Hyperbolic knots : E K are hyperbolic.

23 Example Unlike mapping torus case, geometry of closed braid and the Nielsen-Thurston classification of braids are not in one-to-one correspondence.

24 Example Unlike mapping torus case, geometry of closed braid and the Nielsen-Thurston classification of braids are not in one-to-one correspondence. (a) (b) (c) (a):periodic (b):pseudo-anosov (c):reducible

25 Example Unlike mapping torus case, geometry of closed braid and the Nielsen-Thurston classification of braids are not in one-to-one correspondence. (a) (b) (c) (a):periodic (b):pseudo-anosov (c):reducible Closures of these braids are (2, 3)-torus knot.

26 Correspondence theorem Theorem[Ito] β B n and suppose K = β is knot and [β] D 2 I β is periodic K is a torus knot. I ϕ is reducible K is a satellite knot. I ϕ is pseudo-anosov K is a hyperbolic knot.

27 Correspondence theorem Theorem[Ito] β B n and suppose K = β is knot and [β] D 2 I β is periodic K is a torus knot. I ϕ is reducible K is a satellite knot. I ϕ is pseudo-anosov K is a hyperbolic knot. The Nielsen-Thurston classification and geometry of knot complements are in one-to-one correspondence under the condition [β] D 2.

28 Remark In this Theorem, a condition [β] D 2 is best-possible. A braid β = σ 1 σ2 3σ2 1 knot, and [β] D = 1. is pseudo-anosov, but its closure is satellite