Counting Braids and Laminations Vincent Jug cole des Mines de Paris - PowerPoint PPT Presentation

Counting Braids and Laminations Vincent Jugé École des Mines de Paris & Université Paris Diderot (LIAFA) 10/06/2015 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Contents Braids and Diagrams 1 Braid Groups Complexity of a Braid Band Laminations 2 3 Radial Laminations 4 Conclusion Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Groups What are braids? 1 Intertwined strands Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Groups What are braids? 1 Intertwined strands 2 Isotopy group of braid diagrams “ “ “ ˆ 4 4 3 “ 3 3 2 “ 2 2 1 1 1 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Groups What are braids? 1 Intertwined strands 2 Isotopy group of braid diagrams “ “ “ ˆ 4 4 3 “ 3 3 2 “ 2 2 1 σ 1 σ 2 σ 1 σ 2 σ 1 σ 2 1 1 σ 1 σ 3 σ 3 σ 1 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Groups What are braids? 1 Intertwined strands 2 Isotopy group of braid diagrams 3 Isotopy group of homeomorphisms of C Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Groups What are braids? 1 Intertwined strands 2 Isotopy group of braid diagrams 3 Isotopy group of homeomorphisms of C that fix B D pointwise B D Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Groups What are braids? 1 Intertwined strands 2 Isotopy group of braid diagrams 3 Isotopy group of homeomorphisms of C that fix B D pointwise and let P n globally invariant B D P 5 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Groups What are braids? 1 Intertwined strands 2 Isotopy group of braid diagrams 3 Isotopy group of homeomorphisms of C that fix B D pointwise and let P n globally invariant : B n “ Hom p C , P n Ø P n , Id B D q Hom 0 p C , P n Ø P n , Id B D q . B D P 5 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Groups What are braids? 1 Intertwined strands 2 Isotopy group of braid diagrams 3 Isotopy group of homeomorphisms of C that fix B D pointwise and let P n globally invariant : B n “ Hom p C , P n Ø P n , Id B D q Hom 0 p C , P n Ø P n , Id B D q . 4 Finitely presented group B n “ x σ 1 , . . . , σ n ´ 1 | σ i σ i ` 1 σ i “ σ i ` 1 σ i σ i ` 1 , σ i σ j “ σ j σ i if ě i ` 2 y . σ i : Artin Generators Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Groups What are braids? 1 Intertwined strands 2 Isotopy group of braid diagrams 3 Isotopy group of homeomorphisms of C that fix B D pointwise and let P n globally invariant : B n “ Hom p C , P n Ø P n , Id B D q Hom 0 p C , P n Ø P n , Id B D q . 4 Finitely presented group B n “ x σ 1 , . . . , σ n ´ 1 | σ i σ i ` 1 σ i “ σ i ` 1 σ i σ i ` 1 , σ i σ j “ σ j σ i if ě i ` 2 y . σ i : Artin Generators Coxeter Group: S n “ x σ 1 , . . . , σ n ´ 1 | σ 2 i “ 1 , σ i σ i ` 1 σ i “ σ i ` 1 σ i σ i ` 1 , σ i σ j “ σ j σ i si j ě i ` 2 y . Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #1: a braid with lots of crossings Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #1: a braid with lots of crossings simple complex complex ? Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #1: a braid with lots of crossings simple complex complex ? } α } “ minimal number of crossings Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #1: a braid with lots of crossings simple complex complex ? } α } “ minimal number of crossings } α } “ distance to ε in a Cayley graph: } α ¨ β } ď } α } ` } β } Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #1: a braid with lots of crossings simple complex complex ? } α } “ minimal number of crossings } α } “ distance to ε in a Cayley graph: } α ¨ β } ď } α } ` } β } Computing } α } : very hard Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #1: a braid with lots of crossings simple complex complex ? } α } “ minimal number of crossings } α } “ distance to ε in a Cayley graph: } α ¨ β } ď } α } ` } β } Computing } α } : very hard ( easy up to a multiplicative factor n ! ) Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #1: a braid with lots of crossings simple complex complex ? } α } “ minimal number of crossings } α } “ distance to ε in a Cayley graph: } α ¨ β } ď } α } ` } β } Computing } α } : very hard ( easy up to a multiplicative factor n ! ) Computing N p k q “ # t α : } α } “ k u : seems very hard Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #2: distance to ε in another Cayley graph Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #2: distance to ε in another Cayley graph generated by simple braids Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #2: distance to ε in another Cayley graph generated by simple braids } α } 2 “ distance to ε in a Cayley graph: } α ¨ β } 2 ď } α } 2 ` } β } 2 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #2: distance to ε in another Cayley graph generated by simple braids } α } 2 “ distance to ε in a Cayley graph: } α ¨ β } 2 ď } α } 2 ` } β } 2 Computing } α } 2 : easy Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #2: distance to ε in another Cayley graph generated by simple braids } α } 2 “ distance to ε in a Cayley graph: } α ¨ β } 2 ď } α } 2 ` } β } 2 Computing } α } 2 : easy Computing N p k q “ # t α : } α } 2 “ k u : easy 2 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Complexity of a Braid What is a complex braid? Idea #2: distance to ε in another Cayley graph generated by simple braids } α } 2 “ distance to ε in a Cayley graph: } α ¨ β } 2 ď } α } 2 ` } β } 2 Computing } α } 2 : easy 2 z k is rational) Computing N p k q k ě 0 N p k q “ # t α : } α } 2 “ k u : easy ( ř 2 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Contents Braids and Diagrams 1 2 Band Laminations What are Band Laminations? Laminations and Complexity 3 Radial Laminations 4 Conclusion Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Band Laminations? Trivial band lamination: Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Band Laminations? Non-trivial band lamination: Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Band Laminations? Braid acting on a band lamination: σ 2 0 1 2 3 4 5 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Band Laminations? Braid acting on a band lamination: Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Band Laminations? Braid acting on a band lamination: Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Band Laminations? Braid acting on a band lamination: Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Band Laminations? Braid acting on a band lamination: 0 1 2 3 4 5 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Band Laminations? Braid acting on a band lamination: 1 2 σ ´ 1 2 3 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Acting on a Band Lamination Braid ” Band lamination B n acts faithfully and transitively on L b n : B n “ t n -strand braids u L b n “ t band laminations with n holes u Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Acting on a Band Lamination Braid ” Band lamination B n acts faithfully and transitively on L b n : L b B n ” n α p L b Ñ ε q α B n “ t n -strand braids u L b n “ t band laminations with n holes u L b ε “ trivial band lamination Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations