Handlebodies, Artin–Tits and HOMFLYPT

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Handlebodies, Artin–Tits and HOMFLYPT

Or: All I know about Artin–Tits groups; and a filler for the remaining59minutes Daniel Tubbenhauer

Joint with David Rose

March 2019

Daniel Tubbenhauer Handlebodies, Artin–Tits and HOMFLYPT March 2019 1 / 15

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braids in a3-ball𝒟³

However, this “naive” approach fails for most3-manifolds. Why? Because I do not know what Hecke/Soergel analog

to use for an arbitrary3-manifold.

Today. I explain what we can do.

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braids in a3-ball𝒟³ 𝒷

links in a3-ball𝒟³

Alexander’s theorem

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𝒷

𝒸 𝒷

𝒸

=

extra relations for braids

Markov’s theorem

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

algebraic way to study

combine

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Type A He- cke algebra

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Satisfies braid relations

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

Invariant of links in𝒟³

combine

HOMFLYPT polynomial

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

combine

Type A He- cke category

HOMFLYPT polynomial

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

combine

Braid invariant

HOMFLYPT polynomial

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

combine

Braid invariant

Markov invariant

Markov 2-trace

HOMFLYPT polynomial

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

combine

Braid invariant

Markov invariant

Markov 2-trace

combine

HOMFLYPT homology

HOMFLYPT polynomial

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braids in a3-mfdℳ³

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braids in a3-mfdℳ³ 𝒷

links in a3-mfdℳ³

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Type ? He- cke algebra

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Satisfies braid relations?

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

Invariant of links inℳ³

combine

“HOMFLYPT polynomial”

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

combine

Type ? He- cke category

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

combine

Braid invariant

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

combine

Braid invariant

Markov invariant

Markov 2-trace?

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

combine

Braid invariant

Markov invariant

Markov 2-trace?

combine

“HOMFLYPT homology”

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

combine

Braid invariant

Markov invariant

Markov 2-trace?

combine

However, this “naive” approach fails for most3-manifolds.

Why? Because I do not know what Hecke/Soergel analog to use for an arbitrary3-manifold.

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

combine

Braid invariant

Markov invariant

Markov 2-trace?

combine

However, this “naive” approach fails for most3-manifolds.

Why? Because I do not know what Hecke/Soergel analog to use for an arbitrary3-manifold.

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1 Links and braids in handlebodies Braid diagrams

Links in handlebodies

2 Some “low-genus-coincidences”

The ball

The torus and the double torus

3 Arbitrary genus What we should do What we can do

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LetBr(g, n)be the group defined as follows.

Generators. Braid and twist generators

𝒷i!

1 1

g g

1 1

i+1 i

i i+1

n n

... ... ... & 𝓉i!

1 1

g g

i 1

2 2

n n ...

...

Relations. Reidemeister braid relations , type C relations and special relations,e.g.

=

𝒷1𝓉2𝒷1𝓉2=𝓉2𝒷1𝓉2𝒷1

& =

(𝒷1𝓉2𝒷⁻¹1 )𝓉3=𝓉3(𝒷1𝓉2𝒷⁻¹1 ) Involves three players and inverses!

Example.

The “full wrap”.

=

Fact (typeA embedding).

Br(g, n)is a subgroup of the usual braid groupℬr(g+n).

= 7→ =

A visualization exercise.

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𝒷i!

1 1

g g

1 1

i+1 i

i i+1

n n

... ... ... & 𝓉i!

1 1

g g

i 1

2 2

n n ...

...

=

& =

Example.

=

= 7→ =

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𝒷i!

1 1

g g

1 1

i+1 i

i i+1

n n

... ... ... & 𝓉i!

1 1

g g

i 1

2 2

n n ...

...

=

& =

Example.

=

= 7→ =

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𝒷i!

1 1

g g

1 1

i+1 i

i i+1

n n

... ... ... & 𝓉i!

1 1

g g

i 1

2 2

n n ...

...

=

& =

Example.

=

= 7→ =

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The groupℬr(g, n)of braid in ag-times punctures disk𝒟g²×[0,1]:

Two types of braidings, the usual ones and “winding around cores”,e.g.

𝒟3²×[0,1]

&

𝒟3²×[0,1]

Theorem (H¨aring-Oldenburg–Lambropoulou∼2002, Vershinin∼1998). The map

7→

is an isomorphism of groupsBr(g, n)→ℬr(g, n). From this perspective the type A embedding

is just shrinking holes to points!

shrink

Note.

For the proof it is crucial that𝒟g² and the boundary points of the braids• are only defined up to isotopy,e.g.

•

• 𝒟3²

∼= ^• ^•

𝒟3²

⇒one can always “conjugate cores to the left”. This is useful to defineℬr(g,∞).

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𝒟3²×[0,1]

&

𝒟3²×[0,1]

Theorem (H¨aring-Oldenburg–Lambropoulou∼2002, Vershinin∼1998).

The map

7→

is an isomorphism of groupsBr(g, n)→ℬr(g, n).

From this perspective the type A embedding is just shrinking holes to points!

shrink

Note.

•

• 𝒟3²

∼= ^• ^•

𝒟3²

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𝒟3²×[0,1]

&

𝒟3²×[0,1]

7→

is an isomorphism of groupsBr(g, n)→ℬr(g, n).

From this perspective the type A embedding is just shrinking holes to points!

shrink

Note.

•

• 𝒟3²

∼= ^• ^•

𝒟3²

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𝒟3²×[0,1]

&

𝒟3²×[0,1]

7→

is an isomorphism of groupsBr(g, n)→ℬr(g, n). From this perspective the type A embedding

is just shrinking holes to points!

shrink

Note.

•

• 𝒟3²

∼= • •

𝒟3²

⇒one can always “conjugate cores to the left”.

This is useful to defineℬr(g,∞).

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The Alexander closure onℬr(g,∞)is given by merging core strands at infinity.

wrong closure correct closure

This is different from the ^classical Alexander closure.

Theorem (Lambropoulou∼1993).

For any link𝓁in the genusghandlebodyℋg there is a braid inℬr(g,∞)whose (correct!) closure is isotopic to𝓁.

Fact.

ℋg is given by a complement in the3-sphere𝒮³ by an open tubular neighborhood of the embedded graph obtained

by gluingg+ 1unknotted “core” edges to two vertices.

𝒮3

the3-ballℋ0=𝒟³

𝒮3

a torusℋ1

𝒮3

ℋ2

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Fact.

𝒮3

a torusℋ1

𝒮3

ℋ2

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Fact.

𝒮3

a torusℋ1

𝒮3

ℋ2

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The Markov moves onℬr(g,∞)are conjugation and stabilization.

Conjugation.

𝒷∼𝓈𝒷𝓈⁻¹

for𝒷∈ℬr(g, n),𝓈∈ h𝒷1, . . . ,𝒷n−1i ⇐⇒

... ...

n

... ...

n

𝒷 ∼

...

n ...

...

n

𝒷 𝓈

𝓈^-1

Stabilization.

(𝒸↑)𝒷n(𝒷↑)

∼𝒸𝒷∼(𝒸↑)𝒷⁻n¹(𝒷↑) for𝒷,𝒸∈ℬr(g, n),

⇐⇒

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

They are weaker than the ^classical Markov moves.

Theorem (H¨aring-Oldenburg–Lambropoulou∼2002). Two links inℋg are equivalent if and only if

they are equal inℬr(g,∞)up to conjugation and stabilization. Example.

𝒷

wrong closure

𝒷

correct closure

not stuck stuck

The upshot.

Together with Alexander’s theorem, this gives a way to algebraically study

links inℋg.

Let me explain what we can do.

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Conjugation.

for𝒷∈ℬr(g, n),𝓈∈ h𝒷1, . . . ,𝒷n−1i ⇐⇒

... ...

n

... ...

n

𝒷 ∼

...

n ...

...

n

𝒷 𝓈

𝓈^-1

Stabilization.

⇐⇒

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

Theorem (H¨aring-Oldenburg–Lambropoulou∼2002).

Two links inℋg are equivalent if and only if

they are equal inℬr(g,∞)up to conjugation and stabilization.

Example.

𝒷

wrong closure

𝒷

correct closure

not stuck stuck

The upshot.

links inℋg.

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Conjugation.

for𝒷∈ℬr(g, n),𝓈∈ h𝒷1, . . . ,𝒷n−1i ⇐⇒

... ...

n

... ...

n

𝒷 ∼

...

n ...

...

n

𝒷 𝓈

𝓈^-1

Stabilization.

⇐⇒

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

Theorem (H¨aring-Oldenburg–Lambropoulou∼2002).

Two links inℋg are equivalent if and only if

they are equal inℬr(g,∞)up to conjugation and stabilization.

Example.

𝒷

wrong closure

𝒷

correct closure

The upshot.

links inℋg.

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Conjugation.

for𝒷∈ℬr(g, n),𝓈∈ h𝒷1, . . . ,𝒷n−1i ⇐⇒

... ...

n

... ...

n

𝒷 ∼

...

n ...

...

n

𝒷 𝓈

𝓈^-1

Stabilization.

⇐⇒

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

𝒷

wrong closure

𝒷

correct closure

not stuck stuck

The upshot.

links inℋg.

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Conjugation.

for𝒷∈ℬr(g, n),𝓈∈ h𝒷1, . . . ,𝒷n−1i ⇐⇒

... ...

n

... ...

n

𝒷 ∼

...

n ...

...

n

𝒷 𝓈

𝓈^-1

Stabilization.

⇐⇒

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

𝒷

wrong closure

𝒷

correct closure

not stuck stuck

The upshot.

links inℋg.

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LetΓbe a Coxeter graph.

Artin∼1925, Tits∼1961++. The Artin–Tits group and its Coxeter group quotient are given by generators-relations:

AT(Γ) =h𝒷i| · · ·𝒷i𝒷j𝒷i

| {z }

mij factors

=· · ·𝒷j𝒷i𝒷j

| {z }

mijfactors

i

W(Γ) =hσi |σ_i²= 1,· · ·σiσjσi

| {z }

mijfactors

=· · ·σjσiσj

| {z }

mijfactors

i

Artin–Tits groups ^generalize classical braid groups, Coxeter groups ^generalize polyhedron groups.

My failure. What I would like to understand, but I do not. Artin–Tits groups come in four main flavors.

Question: Why are these special? What happens in general type?

Many open problems,e.g. the

word problem. Flavor one. Finite

and affine types ^helps

Flavor two. Con-

figuration spaces ^helps

Flavor three. Map- ping class groups

helps

Flavor four. Right angled groups

helps

Artin–Tits (braid) groups

Vanilla flavor. ?????.

?

A different idea for today:

What can Artin–Tits groups tell you about flavor two? Jones ∼1987, Geck–Lambropoulou∼1997, Gomi∼2006

In finite type: Markov trace on the Hecke algebras .

Khovanov∼2005, Rouquier ∼2012, Webster–Williamson∼2009; categorification. In finite type: Hochschild homology on complexes of the Hecke category .

Corollary.

HOMFLYPT polynomial/homology for links in ????

q=Hecke parameter ; t=homological parameter ; a=trace parameter .

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| {z }

mij factors

| {z }

mijfactors

i

| {z }

mijfactors

=· · ·σjσiσj

| {z }

mijfactors

i

My failure. What I would like to understand, but I do not.

Artin–Tits groups come in four main flavors.

word problem.

Flavor one. Finite

Flavor two. Con-

helps

?

What can Artin–Tits groups tell you about flavor two?

Jones ∼1987, Geck–Lambropoulou∼1997, Gomi∼2006 In finite type: Markov trace on the Hecke algebras .

Corollary.

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| {z }

mij factors

| {z }

mijfactors

i

| {z }

mijfactors

=· · ·σjσiσj

| {z }

mijfactors

i

Flavor two. Con-

helps

?

Corollary.

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| {z }

mij factors

| {z }

mijfactors

i

| {z }

mijfactors

=· · ·σjσiσj

| {z }

mijfactors

i

Flavor two. Con-

helps

?

Khovanov∼2005, Rouquier ∼2012, Webster–Williamson∼2009; categorification.

In finite type: Hochschild homology on complexes of the Hecke category .

Corollary.

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| {z }

mij factors

| {z }

mijfactors

i

| {z }

mijfactors

=· · ·σjσiσj

| {z }

mijfactors

i

Flavor two. Con-

helps

?

Khovanov∼2005, Rouquier ∼2012, Webster–Williamson∼2009; categorification.

In finite type: Hochschild homology on complexes of the Hecke category .

Corollary.

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cos(π/3) on a line:

type A_n−1: 1 2 . . . n−2 n−1

The classical case. Consider the map

βi 7→

1 1

i+1 i

i i+1

n n

... ... braid rel.: =

Artin∼1925. This gives an isomorphism of groupsAT(A_n−1)−→^∼⁼ ℬr(0, n).

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cos(π/4) on a line:

type Cn: 0 ⁴ 1 2 . . . n−1 n

The semi-classical case. Consider the map

β07→

1 1

2 2

n n

... & βi7→

1 1

i+1 i

i i+1

n n

... ... braid rel.: =

Brieskorn ∼1973. This gives an isomorphism of groupsAT(Cn)−→^∼⁼ ℬr(1, n).