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

1

Braids and Diagrams Braid Groups Complexity of a Braid

2

Band Laminations

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 “ “ “

1 2 3 4

“

1 2 3 4 1 2 3

“

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 “ “ “

1 2 3 4

“

1 2 3 4

σ1 σ3 σ3 σ1

1 2 3

“ σ1 σ2 σ1 σ2 σ1 σ2

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 BD pointwise

BD

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 BD pointwise and

let Pn globally invariant BD P5

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 BD pointwise and

let Pn globally invariant: Bn “ HompC,PnØPn,IdBDq

Hom0pC,PnØPn,IdBDq.

BD P5

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 BD pointwise and

let Pn globally invariant: Bn “ HompC,PnØPn,IdBDq

Hom0pC,PnØPn,IdBDq.

4 Finitely presented group

Bn “ xσ1, . . . , σn´1 | σiσi`1σi “ σi`1σiσi`1, σiσj “ σjσi if ě i ` 2y. σi : 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 BD pointwise and

let Pn globally invariant: Bn “ HompC,PnØPn,IdBDq

Hom0pC,PnØPn,IdBDq.

4 Finitely presented group

Bn “ xσ1, . . . , σn´1 | σiσi`1σi “ σi`1σiσi`1, σiσj “ σjσi if ě i ` 2y. σi : Artin Generators

Coxeter Group:

Sn “ xσ1, . . . , σn´1 | σ2

i “ 1, σiσi`1σi “ σi`1σiσi`1, σiσj “ σjσi si j ě i ` 2y.

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 Npkq “ #tα : }α} “ ku: 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 Npkq

2

“ #tα : }α}2 “ ku: 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 Npkq

2

“ #tα : }α}2 “ ku: easy (ř

kě0 Npkq 2 zk is rational)

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Contents

1

Braids and Diagrams

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:

σ2 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:

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:

σ´1

2

3 2 1 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Acting on a Band Lamination

Braid ” Band lamination

Bn acts faithfully and transitively on Lb

n :

Bn “ tn-strand braidsu Lb

n “ tband laminations with n holesu

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Acting on a Band Lamination

Braid ” Band lamination

Bn acts faithfully and transitively on Lb

n :

Bn ” Lb

n

α Ñ αpLb

εq

Bn “ tn-strand braidsu Lb

n “ tband laminations with n holesu

Lb

ε “ trivial band lamination

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Acting on a Band Lamination

Braid ” Band lamination

Bn acts faithfully and transitively on Lb

n :

Bn ” Lb

n

α Ñ αpLb

εq

Bn “ tn-strand braidsu Lb

n “ tband laminations with n holesu

Lb

ε “ trivial band lamination

ε σ´1

1 σ´1 2 σ3

σ2σ´1

1 σ´1 2 σ3

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #3: a band lamination whose arcs often cross R

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #3: a band lamination whose arcs often cross R Complex braid

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #3: a band lamination whose arcs often cross R Complex braid }α}3 “ cardinality of αpLc

εq X R

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #3: a band lamination whose arcs often cross R Complex braid σ1σ´1

2

}α}3 “ cardinality of αpLc

εq X R

}pσ1σ´1

2 qk}3 « 2k

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #3: a band lamination whose arcs often cross R Complex braid σ1σ´1

2

}α}3 “ cardinality of αpLc

εq X R

}pσ1σ´1

2 qk}3 « 2k: }α ¨ β}3 ę }α}3 ` }β}3

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #3: a band lamination whose arcs often cross R Complex braid σ1σ´1

2

}α}3 “ cardinality of αpLc

εq X R

}pσ1σ´1

2 qk}3 « 2k: }α ¨ β}3 ę }α}3 ` }β}3

Computing }α}3: easy

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #3: a band lamination whose arcs often cross R Complex braid σ1σ´1

2

}α}3 “ cardinality of αpLc

εq X R

}pσ1σ´1

2 qk}3 « 2k: }α ¨ β}3 ę }α}3 ` }β}3

Computing }α}3: easy Computing Npkq

3

“ #tα : }α}3 “ ku: not obvious. . .

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Contents

1

Braids and Diagrams

2

Band Laminations

3

Radial Laminations What are Radial Laminations? Laminations and Complexity Counting Laminations

4

Conclusion

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Radial Laminations?

Trivial radial lamination:

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Radial Laminations?

Non-trivial radial lamination:

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Radial Laminations?

Braid acting on a radial lamination:

σ2 1 2 3 4 5

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Radial Laminations?

Braid acting on a radial lamination:

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Radial Laminations?

Braid acting on a radial lamination:

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Radial Laminations?

Braid acting on a radial lamination:

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Radial Laminations?

Braid acting on a radial lamination:

1 2 3 4 5

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

What are Radial Laminations?

Braid acting on a radial lamination:

σ´1

2

3 2 1 Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Acting on a Radial Lamination

Braid ” Radial lamination

Bn acts faithfully and transitively on Lr

n :

Bn “ tn-strand braidsu Lr

n “ tradial laminations with n holesu

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Acting on a Radial Lamination

Braid ” Radial lamination

Bn acts faithfully and transitively on Lr

n :

Bn ” Lr

n

α Ñ αpLr

εq

Bn “ tn-strand braidsu Lr

n “ tradial laminations with n holesu

Lr

ε “ trivial radial lamination

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Braid Acting on a Radial Lamination

Braid ” Radial lamination

Bn acts faithfully and transitively on Lr

n :

Bn ” Lr

n

α Ñ αpLr

εq

Bn “ tn-strand braidsu Lr

n “ tradial laminations with n holesu

Lr

ε “ trivial radial lamination

ε σ´1

1 σ´1 2 σ3

σ2σ´1

1 σ´1 2 σ3

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #4: a lamination whose ray often crosses Lb

ε

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #4: a lamination whose ray often crosses Lb

ε

Complex braid

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #4: a lamination whose ray often crosses Lb

ε

Complex braid }α}4 “ cardinality of αpLr

εq X Lb ε

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #4: a lamination whose ray often crosses Lb

ε

Complex braid }α}4 “ cardinality of αpLr

εq X Lb ε

Computing Npkq

4

“ #tα : }α}4 “ ku

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

What is a complex braid?

Idea #4: a lamination whose ray often crosses Lb

ε

Complex braid }α}4 “ cardinality of αpLr

εq X Lb ε“ }α´1}3

Computing Npkq

4

“ #tα : }α}4 “ ku “ Npkq

3 : not so hard. . .

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

Why do we have }α}4 “ }α´1}3?

Pull α’s ray tight!

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

Why do we have }α}4 “ }α´1}3?

Pull α’s ray tight! |σ2σ´1

1 pLr εq X Lb ε|

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

Why do we have }α}4 “ }α´1}3?

Pull α’s ray tight! |σ2σ´1

1 pLr εq X Lb ε|

|σ´1

2 pLb εq X σ´1 1 pLr εq|

=

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Laminations and Complexity

Why do we have }α}4 “ }α´1}3?

Pull α’s ray tight! |σ2σ´1

1 pLr εq X Lb ε|

|σ´1

2 pLb εq X σ´1 1 pLr εq|

= |σ1σ´1

2 pLb εq X Lr ε|

=

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting Laminations

How can we count (radial) laminations?

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting Laminations

How can we count (radial) laminations?

1 Identify mirrors Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting Laminations

How can we count (radial) laminations?

1 Identify mirrors and their periscopes Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting Laminations

How can we count (radial) laminations?

1 Identify mirrors and their periscopes and transparent holes Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting Laminations

How can we count (radial) laminations?

1 Identify mirrors and their periscopes and transparent holes 2 Check that the ray is connected! Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting Laminations

How can we count (radial) laminations?

1 Identify mirrors and their periscopes and transparent holes 2 Check that the ray is connected! Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 1 or 2 strands

1-strand braids:

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 1 or 2 strands

1-strand braids: Npkq

4

“ 1k“0

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 1 or 2 strands

1-strand braids: Npkq

4

“ 1k“0 2-strand braids: Npkq

4

“ 1k“1

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 1 or 2 strands

1-strand braids: Npkq

4

“ 1k“0 2-strand braids: Npkq

4

“ 1k“1 ` 2 ¨ 1kP2N`3

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 1 or 2 strands

1-strand braids: Npkq

4

“ 1k“0 2-strand braids: Npkq

4

“ 1k“1 ` 2 ¨ 1kP2N`3

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 3 strands

3-strand braids:

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 3 strands

3-strand braids:

Npkq

4

“ 1k“2 ` 2ϕpk{2 ` 1q ¨ 1kP2N`4

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 3 strands

3-strand braids:

Npkq

4

“ 1k“2 ` 2ϕpk{2 ` 1q ¨ 1kP2N`4 ´ 2 ¨ 1kP4N`6

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 3 strands

3-strand braids:

Npkq

4

“ 1k“2 ` 2ϕpk{2 ` 1q ¨ 1kP2N`4 ´ 2 ¨ 1kP4N`6 ` 4

k{4

ÿ

i“2

ϕpk{2 ` 4 ´ 2iq ¨ 1kP2N`2

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 3 strands

3-strand braids:

Npkq

4

“ 1k“2 ` 2ϕpk{2 ` 1q ¨ 1kP2N`4 ´ 2 ¨ 1kP4N`6 ` 4

k{4

ÿ

i“2

ϕpk{2 ` 4 ´ 2iq ¨ 1kP2N`2 Npkq

4

„ p1kP2N ` 1kP4N`2qk2{π2

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 3 strands

3-strand braids:

Npkq

4

“ 1k“2 ` 2ϕpk{2 ` 1q ¨ 1kP2N`4 ´ 2 ¨ 1kP4N`6 ` 4

k{4

ÿ

i“2

ϕpk{2 ` 4 ´ 2iq ¨ 1kP2N`2 Npkq

4

„ p1kP2N ` 1kP4N`2qk2{π2 ÿ

kě0

Npkq

4 zk

“ 21 ` 2z2 ´ z4 z2p1 ´ z4q ˜ÿ

ně3

ϕpnqz2n ¸ ` z2p1 ´ 3z4q 1 ´ z4

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 3 strands

3-strand braids:

Npkq

4

“ 1k“2 ` 2ϕpk{2 ` 1q ¨ 1kP2N`4 ´ 2 ¨ 1kP4N`6 ` 4

k{4

ÿ

i“2

ϕpk{2 ` 4 ´ 2iq ¨ 1kP2N`2 Npkq

4

„ p1kP2N ` 1kP4N`2qk2{π2 ÿ

kě0

Npkq

4 zk

“ 21 ` 2z2 ´ z4 z2p1 ´ z4q ˜ÿ

ně3

ϕpnqz2n ¸ ` z2p1 ´ 3z4q 1 ´ z4 Typical cases:

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 4 strands or more

n-strand braids:

Npkq

4

‰ 0 ô k P 2N ` n ´ 1

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 4 strands or more

n-strand braids:

Npkq

4

‰ 0 ô k P 2N ` n ´ 1 Ñ Mℓ “ Npn´1`2ℓq

4

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 4 strands or more

n-strand braids:

Npkq

4

‰ 0 ô k P 2N ` n ´ 1 Ñ Mℓ “ Npn´1`2ℓq

4

Mℓ “ Opℓ2n´4q

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 4 strands or more

n-strand braids:

Npkq

4

‰ 0 ô k P 2N ` n ´ 1 Ñ Mℓ “ Npn´1`2ℓq

4

Mℓ “ Opℓ2n´4q ℓn´2 “ OpMℓq

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 4 strands or more

n-strand braids:

Npkq

4

‰ 0 ô k P 2N ` n ´ 1 Ñ Mℓ “ Npn´1`2ℓq

4

Mℓ “ Opℓ2n´4q ℓtp3n´5q{2u “ OpMℓq

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 4 strands or more

n-strand braids:

Npkq

4

‰ 0 ô k P 2N ` n ´ 1 Ñ Mℓ “ Npn´1`2ℓq

4

Mℓ “ Opℓ2n´4q ℓtp3n´5q{2u “ OpMℓq

Conjecture

Mℓ “ Θpℓ2n´4q

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Counting laminations: 4 strands or more

n-strand braids:

Npkq

4

‰ 0 ô k P 2N ` n ´ 1 Ñ Mℓ “ Npn´1`2ℓq

4

Mℓ “ Opℓ2n´4q ℓtp3n´5q{2u “ OpMℓq

Conjecture

Mℓ “ Θpℓ2n´4q Is this permutation cyclic?

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Contents

1

Braids and Diagrams

2

Band Laminations

3

Radial Laminations

4

Conclusion

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Conclusion

Next goals

Prove the conjecture Look at the combinatorial structure of laminations

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations

Conclusion

Next goals

Prove the conjecture Look at the combinatorial structure of laminations

Thank you!

Vincent Jugé (Mines Paris & Paris 7 – LIAFA) Counting Braids and Laminations