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

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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. ?????.

?

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?

A different idea for today:

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

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

What can Artin–Tits groups tell you about flavor two? However, this “naive” approach fails for most3-manifolds. Why? Because I do not know what Hecke/Soergel analog

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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

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov

Invariant of links in𝒟³

combine

HOMFLYPT polynomial

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov

combine

Type A He- cke category

HOMFLYPT polynomial

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov

combine

Braid invariant

HOMFLYPT polynomial

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov

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

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

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov

Invariant of links inℳ³

combine

“HOMFLYPT polynomial”

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov

combine

Type ? He- cke category

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov

combine

Braid invariant

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov

combine

Braid invariant

Markov invariant

Markov 2-trace?

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𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov

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

combine

Braid invariant

Markov invariant

Markov 2-trace?

combine

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

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

combine

Braid invariant

Markov invariant

Markov 2-trace?

combine

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

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 – the classical case Braid diagrams

Links in the 3-ball

2 Links and braids in handlebodies Braid diagrams

Links in handlebodies

3 Some “low-genus-coincidences”

The ball and the torus

The torus and the double torus

4 Arbitrary genus

Braid invariants – some ideas Link invariants – some ideas

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LetBr(n)be the group defined as follows.

Generators. Braid generators

𝒷i!

1 1

i+1 i

i i+1

n n ... ... ...

Relations. Reidemeister braid relations,e.g.

= =

𝒷i𝒷⁻_i¹= 1 =𝒷⁻_i¹𝒷i

& =

𝒷i+1𝒷i𝒷i+1=𝒷i𝒷i+1𝒷i

Example.

“Topology in disguise”.

=

Theorem (Gauß≤1830 ^? , Artin ∼1925.)

Letℬr(n)be the group of braids in a disk𝒟²×[0,1]. The map

7→

𝒟²×[0,1]

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

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𝒷i!

1 1

i+1 i

i i+1

n n ... ... ...

= =

& =

Example.

=

7→

𝒟²×[0,1]

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𝒷i!

1 1

i+1 i

i i+1

n n ... ... ...

= =

& =

Example.

=

7→

𝒟²×[0,1]

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𝒷i!

1 1

i+1 i

i i+1

n n ... ... ...

= =

& =

Example.

=

Theorem (Gauß≤1830 ^? , Artin∼1925.)

Letℬr(n)be the group of braids in a disk𝒟²×[0,1].

The map

7→

𝒟²×[0,1]

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The Alexander closure onℬr(∞)is given by:

This is the classical Alexander closure.

Theorem (Brunn∼1897, Alexander∼1923). For any link𝓁in the3-ball𝒟³ there is a braid inℬr(∞)

whose closure is isotopic to𝓁.

Proof?

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The Alexander closure onℬr(∞)is given by:

This is the classical Alexander closure.

Theorem (Brunn∼1897, Alexander∼1923).

For any link𝓁in the3-ball𝒟³ there is a braid inℬr(∞) whose closure is isotopic to𝓁.

Proof?

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

Conjugation.

... ...

𝒷 ∼

... ...

𝒷 𝒸

𝒸^-1

⇔

... ...

𝒸

𝒷

∼

... ...

𝒷

𝒸

Stabilization.

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

These are the classical Markov moves.

Theorem (Markov∼1936). Two links in𝒟³ are equivalent if and only if they are equal inℬr(∞)up to conjugation and stabilization.

Proof?

The upshot.

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

links in𝒟³.

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Conjugation.

... ...

𝒷 ∼

... ...

𝒷 𝒸

𝒸^-1

⇔

... ...

𝒸

𝒷

∼

... ...

𝒷

𝒸

Stabilization.

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

Theorem (Markov∼1936).

Two links in𝒟³ are equivalent if and only if they are equal inℬr(∞)up to conjugation and stabilization.

Proof?

The upshot.

links in𝒟³.

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Conjugation.

... ...

𝒷 ∼

... ...

𝒷 𝒸

𝒸^-1

⇔

... ...

𝒸

𝒷

∼

... ...

𝒷

𝒸

Stabilization.

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

∼

n n

𝒷 𝒸

Theorem (Markov∼1936).

Two links in𝒟³ are equivalent if and only if they are equal inℬr(∞)up to conjugation and stabilization.

Proof?

The upshot.

links in𝒟³.

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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→

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.

Proof? L-move. 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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Proof? L-move.

Fact.

𝒮3

the3-ballℋ0=𝒟³

𝒮3

a torusℋ1

𝒮3

ℋ2

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Proof? L-move.

Fact.

𝒮3 𝒮3 𝒮3

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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 inℋ^g are equivalent if and only if

they are equal inℬr(g,∞)up to conjugation and stabilization. Proof? L-move.

Example.

𝒷

wrong closure

𝒷

correct closure

not stuck stuck

The upshot.

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 inℋ^g are equivalent if and only if

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

Proof? L-move.

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 inℋ^g are equivalent if and only if

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

Proof? L-move.

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

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

Example.

𝒷

wrong closure

𝒷

correct closure

not stuck stuck

The upshot.

links inℋg.