HOMFLYPT homology for links in handlebodies

(1)

HOMFLYPT homology for links in handlebodies

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

Joint with David Rose

(2)

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.

(3)

braids in a3-ball𝒟³ 𝒷

links in a3-ball𝒟³

Alexander’s theorem

(4)

𝒷

𝒸 𝒷

𝒸

=

extra relations for braids

Markov’s theorem

(5)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

algebraic way to study braids in a3-ball𝒟³

combine

(6)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Type A He- cke algebra

(7)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Satisfies braid relations

(8)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

(9)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

Invariant of links in𝒟³

combine

HOMFLYPT polynomial

(10)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

combine

Type A He- cke category

HOMFLYPT polynomial

(11)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

combine

Braid invariant

HOMFLYPT polynomial

(12)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace

combine

Braid invariant

Markov invariant

Markov 2-trace

HOMFLYPT polynomial

(13)

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

(14)

braids in a3-mfdℳ³

(15)

braids in a3-mfdℳ³ 𝒷

links in a3-mfdℳ³

(16)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

(17)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

algebraic way to study braids in a3-mfdℳ³

combine

(18)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Type ? He- cke algebra

(19)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Satisfies braid relations?

(20)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

(21)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

Invariant of links inℳ³

combine

“HOMFLYPT polynomial”

(22)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

combine

Type ? He- cke category

(23)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

combine

Braid invariant

(24)

𝒷

𝒸 𝒷

𝒸

=

Markov’s theorem

combine

Braid invariant

Markov invariant

Markov trace?

combine

Braid invariant

Markov invariant

Markov 2-trace?

(25)

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”

(26)

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.

(27)

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.

(28)

1 Links and braids in handlebodies Braid diagrams

Links in handlebodies

2 Some “low-genus-coincidences”

The ball and the torus

The torus and the double torus

3 Arbitrary genus

Braid invariants – some ideas Link invariants – some ideas

(29)

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.

(30)

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→ =

(31)

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→ =

(32)

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→ =

(33)

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,∞).

(34)

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

(35)

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

(36)

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,∞).

(37)

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

(38)

Fact.

𝒮3

a torusℋ1

𝒮3

ℋ2

(39)

Fact.

𝒮3

a torusℋ1

𝒮3

ℋ2

(40)

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.

(41)

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.

(42)

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.

(43)

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.

(44)

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.

(45)

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.

(46)

cos(π/3) on a line:

type An−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(An−1)−→^∼⁼ ℬr(0, n).

I will come back to this with more details for general genusg. For the time being: This works quite well!

Jones ∼1987.

Markov trace on the Hecke algebra of type A

two variableq,apolynomial invariant (HOMFLYPT polynomial). q=Hecke parameter ; a=trace parameter .

Khovanov∼2005; categorification.

Hochschild homology on complexes of the Hecke category of type A

“three variableq,t,ahomological invariant” (HOMFLYPT homology). q=Hecke parameter ; t=homological parameter ; a=Hochschild parameter .

(47)

type An−1: 1 2 . . . n−2 n−1

βi 7→

1 1

i+1 i

i i+1

n n

... ... braid rel.: =

I will come back to this with more details for general genusg.

For the time being: This works quite well!

Jones ∼1987.

two variableq,apolynomial invariant (HOMFLYPT polynomial).

q=Hecke parameter ; a=trace parameter .

Khovanov∼2005; categorification.

(48)

type An−1: 1 2 . . . n−2 n−1

βi 7→

1 1

i+1 i

i i+1

n n

... ... braid rel.: =

Jones ∼1987.

q=Hecke parameter ; a=trace parameter . Khovanov∼2005; categorification.

“three variableq,t,ahomological invariant” (HOMFLYPT homology).

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

(49)

cos(π/3) on a circle.

typeA˜n−1: 0

1 2 . . . n−2 n−1

Affine adds genus. Consider the map

β07→

n

1 2

n

1 2

...

& βi7→

1 1

i+1 i

i i+1

n n

... ... braid rel.: =

tom Dieck ∼1998. (Earlier reference?) This gives an isomorphism of groups Z nAT(˜An−1)−→^∼⁼ ℬr(1, n).

Orellana–Ram∼2004. (Earlier reference?)

Markov trace on the Hecke algebra of typeA˜

???; categorification.

Hochschild homology on complexes of the Hecke category of typeA˜

Fact. One can recover the (missing) generator ofZif one works with extended affine type A.

“extended, extra generator”7→

n

1 2

1 2 n

... ...

n

1 2

1 2 n

... ...

and

n

1 2

n

1 2

... ...

give

1 1

2 2

n n ... ...

(50)

typeA˜n−1: 0

1 2 . . . n−2 n−1

β07→

n

1 2

n

1 2

...

& βi7→

1 1

i+1 i

i i+1

n n

... ... braid rel.: =

n

1 2

1 2 n

... ...

n

1 2

1 2 n

... ...

and

n

1 2

n

1 2

... ...

give

1 1

2 2

n n ... ...

(51)

typeA˜n−1: 0

1 2 . . . n−2 n−1

β07→

n

1 2

n

1 2

...

& βi7→

1 1

i+1 i

i i+1

n n

... ... braid rel.: =

“three variableq,t,ahomological invariant” (HOMFLYPT homology).

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

n

1 2

1 2 n

... ...

n

1 2

1 2 n

... ...

and

n

1 2

n

1 2

... ...

give

1 1

2 2

n n ... ...

(52)

typeA˜n−1: 0

1 2 . . . n−2 n−1

β07→

n

1 2

n

1 2

...

& βi7→

1 1

i+1 i

i i+1

n n

... ... braid rel.: =

n

1 2

1 2 n

...

n

1 2

1 2 n

...

and

n

1 2

n

1 2

...

give

1 1

2 2

n n ...

...

(53)

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).

Geck–Lambropoulou∼1997.

Markov trace on the Hecke algebra of type C

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

Hochschild homology on complexes of the Hecke category of type C

Fact. (Not true in typeA.)

There is a whole infinite family of Markov traces, one for each choice of a value for essential unlinks.

!extra parameter and !extra parameter etc.

However, I only know the categorification of one of these. Fact. (Not true in typeA.)

There is also a second Hecke parameter, which we do not know how to categorify yet.

(54)

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).

Geck–Lambropoulou∼1997.

Markov trace on the Hecke algebra of type C

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

Hochschild homology on complexes of the Hecke category of type C

Fact. (Not true in typeA.)

There is a whole infinite family of Markov traces, one for each choice of a value for essential unlinks.

!extra parameter and !extra parameter etc.

However, I only know the categorification of one of these. Fact. (Not true in typeA.)

There is also a second Hecke parameter, which we do not know how to categorify yet.