Categorication of 1 (and of the Alexander polynomial)

Louis-Hadrien Robert Emmanuel Wagner

6th SwissMAP General Meeting HOMFLY-PT polynomial

à ! à ! à ! 1 1 aP a− P (q q− )P , − = − 1 a a− P(unknot) 1 or P(unknot) − . = = q q 1 − − HOMFLY-PT polynomial

à ! à ! à ! 1 1 aP a− P (q q− )P , − = − 1 a a− P(unknot) 1 or P(unknot) − . = = q q 1 − − N slN -invariants a q =

General N: ReshetikhinTuraev sl -invariants (U (sl )), Ï N q N N 2: Jones polynomial, Ï = N 1: Trivial invariant P1 (1 on every link), Ï = N 0: Alexander polynomial. Ï = Jones Khovanov homology Khovanov, '00 ↔ slN -invariant (N 2) slN -homology KhovanovRozansky '08 ≥ ↔

HOMFLY-PT Triply graded homology KhovanovRozansky, '06 ↔

Alexander Knot Floer homology OzsváthSzabó, '04 ↔

Categorication

Philosophy Promote numerical invariants to (graded) vector space valued invariants.

χ de Rham cohomology de Rahm, '31 ↔ Categorication

Philosophy Promote numerical invariants to (graded) vector space valued invariants.

χ de Rham cohomology de Rahm, '31 ↔

Jones Khovanov homology Khovanov, '00 ↔ slN -invariant (N 2) slN -homology KhovanovRozansky '08 ≥ ↔

HOMFLY-PT Triply graded homology KhovanovRozansky, '06 ↔

Alexander Knot Floer homology OzsváthSzabó, '04 ↔ Categorication

Philosophy Promote numerical invariants to (graded) vector space valued invariants.

χ de Rham cohomology de Rahm, '31 ↔

Jones Khovanov homology Khovanov, '00 ↔ slN -invariant (N 2) slN -homology KhovanovRozansky '08 ≥ ↔ Rasmussen, '15 ⇑ HOMFLY-PT Triply graded homology KhovanovRozansky, '06 ↔ ? DuneldGukovRasmussen, '06 ⇓ Alexander Knot Floer homology OzsváthSzabó, '04 ↔ Today

The gl1 link invariant P1 gl1-homology R.Wagner, '18. Ï ↔ Ã ! Ã ! Ã ! 1 1 qP1 q− P1 (q q− )P1 − = −

The Alexander polynomial ∆ gl0-homology R.Wagner, '19. Ï ↔ Ã ! Ã ! Ã ! 1 ∆ ∆ (q q− )∆ − = − gl1 invariant

Link diagram 1 -lin. comb. of plane graphs Z[q,q− ]

1 2 q− q− −

1 2 q+ q+ −

plane graph element of 1 N[q,q− ] 2 ¡ 1¢#V (Γ)/ #V (Γ)/2 Γ q q− [2] . + = 4 5 q− q− −

3 4 5 6 q− q− q− q− − −

4 5 q− q− −

3 3 4 2 5 6 1 q− [2] 3q− [2] 3q− [2] q− = − + −

gl1 invariant  Example 4 5 q− q− −

4 5 6 q− q− q− − −

4 5 q− q− −

3 3 4 2 5 6 1 q− [2] 3q− [2] 3q− [2] q− = − + −

gl1 invariant  Example

3 q− 3 3 4 2 5 6 1 q− [2] 3q− [2] 3q− [2] q− = − + −

gl1 invariant  Example

4 5 q− q− −

3 4 5 6 q− q− q− q− − −

4 5 q− q− − gl1 invariant  Example

4 5 q− q− −

3 4 5 6 q− q− q− q− − −

4 5 q− q− −

3 3 4 2 5 6 1 q− [2] 3q− [2] 3q− [2] q− = − + − Planar (vinyl) graph graded vector space dimension [2]#V (Γ)/2 graded linear map

gl1-homology

Braid closure diagram hypercube of plane graphs graphs (with shifts)

{ 2} { 1} { 2} + −→ − −

{ 2} { 1} + + gl1-homology

Braid closure diagram hypercube of plane graphs graphs (with shifts)

{ 2} { 1} { 2} + −→ − −

{ 2} { 1} + +

Planar (vinyl) graph graded vector space dimension [2]#V (Γ)/2 graded linear map 3 3 4 2 5 6 q− [2] 3q− [2] 3q− [2] q− − + −

gl1-homology  Example

{ 4} { 5} − −

{ 3} { 4} { 5} { 6} − − − −

{ 4} { 5} − − gl1-homology  Example

{ 4} { 5} − −

{ 3} { 4} { 5} { 6} − − − −

{ 4} { 5} − −

3 3 4 2 5 6 q− [2] 3q− [2] 3q− [2] q− − + − k Y #{ in Ci } Xi i 1 τ(d,c) =Y = (Xi Xj ) − Ci Cj

Dot conguration d .

M D(Γ) Q. = d Multiplication µ on D(Γ). Coloring c (C1,C2,..., Ck ) F = Γ i Ci . =

Vinyl graph vector space

Vinyl graph Γ index k. k Y #{ in Ci } Xi i 1 τ(d,c) =Y = (Xi Xj ) − Ci Cj

M D(Γ) Q. = d Multiplication µ on D(Γ). Coloring c (C1,C2,..., Ck ) F = Γ i Ci . =

Vinyl graph vector space

Vinyl graph Γ index k. Dot conguration d . k Y #{ in Ci } Xi i 1 τ(d,c) =Y = (Xi Xj ) − Ci Cj

Multiplication µ on D(Γ). Coloring c (C1,C2,..., Ck ) F = Γ i Ci . =

Vinyl graph vector space

Vinyl graph Γ index k. Dot conguration d .

M D(Γ) Q. = d k Y #{ in Ci } Xi i 1 τ(d,c) =Y = (Xi Xj ) − Ci Cj

Coloring c (C1,C2,..., Ck ) F = Γ i Ci . =

Vinyl graph vector space

Vinyl graph Γ index k. Dot conguration d .

M D(Γ) Q. = d Multiplication µ on D(Γ). k Y #{ in Ci } Xi i 1 τ(d,c) =Y = (Xi Xj ) − Ci Cj

Vinyl graph vector space

Vinyl graph Γ index k. Dot conguration d .

M D(Γ) Q. = d Multiplication µ on D(Γ). Coloring c (C1,C2,..., Ck ) F = Γ i Ci . = Vinyl graph vector space

Vinyl graph Γ index k. Dot conguration d .

M D(Γ) Q. = d Multiplication µ on D(Γ). Coloring c (C1,C2,..., Ck ) F = Γ i Ci . =

k Y #{ in Ci } Xi 2 2 2 i 1 X1 X2 X3X4 τ(d,c) =Y 3 2 = (Xi Xj ) = (X1 X2) (X3 X4) (X1 X4)(X2 X3) − − − − − Ci Cj Vinyl graph vector space

Vinyl graph Γ index k. Dot conguration d .

M D(Γ) Q. = d Multiplication µ on D(Γ). Coloring c (C1,C2,..., Ck ) F = Γ i Ci . =

k Y #{ in Ci } Xi 2 2 2 i 1 X1 X2 X3X4 τ(d,c) =Y 3 − 2 = (Xi Xj ) = (X1 X2) (X3 X4) (X1 X4)(X2 X3) − − − − − Ci Cj Vinyl graph vector space

Vinyl graph Γ index k. Dot conguration d .

M D(Γ) Q. = d Multiplication µ on D(Γ). Coloring c (C1,C2,..., Ck ) F = Γ i Ci . =

k Y #{ in Ci } Xi 2 2 2 i 1 X X1 X2 X3X4 τ(d,c) =Y τ(d) τ(d,c) 3 2 = (Xi Xj ) = c col(Γ) = (X1 X2) (X3 X4) (X1 X4)(X2 X3) − ∈ − − − − Ci Cj Vinyl graph vector space

Proposition (R.Wagner, '17 )

S For any dot conguration d, τ(d) Q[X1,...,Xk ] k . ∈

S1(Γ) D(Γ)/ker(τ µ( , )X 0). = ◦ •7→ Theorem (R.Wagner, '18 )

#V (Γ)/2 For any vinyl graph Γ, dimq S1(Γ) [2] . = Theorem (R.Wagner '18)

1. These maps in the attening of the hypercube produces a chain complex. Its homology, denoted is a link invariant Hgl1 which categories P1. 2. There is a spectral sequence from the triply graded homology to . Hgl1

linear maps

³ ´ ³ ´ ³ ´ ³ ´ : S1 S1 : S1 S1 → → 7→ 7→ − linear maps

³ ´ ³ ´ ³ ´ ³ ´ : S1 S1 : S1 S1 → → 7→ 7→ −

Theorem (R.Wagner '18)

1. These maps in the attening of the hypercube produces a chain complex. Its homology, denoted is a link invariant Hgl1 which categories P1. 2. There is a spectral sequence from the triply graded homology to . Hgl1 Do I have time?

Examples

4 2 1. Trefoil: the Poincaré polynomial is 1 q− (t t ). + + 2. Hopf link: the Poincaré polynomial is 1 q2(1 t). + + Examples

4 2 1. Trefoil: the Poincaré polynomial is 1 q− (t t ). + + 2. Hopf link: the Poincaré polynomial is 1 q2(1 t). + +

Do I have time? Alexander polynomial

Marked ( ) braid closure 1 -lin. comb. of marked plane graphs ? Z[q,q− ]

1 q− −

1 q+ −

Marked plane graph element of 1 N[q,q− ] Γ complicated (comes from Uq(gl(1 1)) mod). | − 2 2 2 1 2 q 1 q− [2] 3q− [2] 3q− 0 − + = − +

Alexander polynomial  Example

1 2 q− q− − ? ?

1 2 3 q− q− q− − − ? ? ? ? ?

1 2 q− q− − ? ? Alexander polynomial  Example

1 2 q− q− − ? ?

1 2 3 q− q− q− − − ? ? ? ? ?

1 2 q− q− − ? ?

2 2 2 1 2 q 1 q− [2] 3q− [2] 3q− 0 − + = − + S00(Γ ) S1(Γ) at least k 1 at ? { k 1} ? ⊆ = 〈 − 〉 − + induced by S1.

Theorem (R.Wagner, '19)

For any right-marked vinyl graph Γ?, dimq S00(Γ?) is the expected graded dimension.

gl0-homology

Same hypercube with a dierent functor. Theorem (R.Wagner, '19)

For any right-marked vinyl graph Γ?, dimq S00(Γ?) is the expected graded dimension.

gl0-homology

Same hypercube with a dierent functor.

S00(Γ ) S1(Γ) at least k 1 at ? { k 1} ? ⊆ = 〈 − 〉 − + induced by S1. gl0-homology

Same hypercube with a dierent functor.

S00(Γ ) S1(Γ) at least k 1 at ? { k 1} ? ⊆ = 〈 − 〉 − + induced by S1.

Theorem (R.Wagner, '19)

For any right-marked vinyl graph Γ?, dimq S00(Γ?) is the expected graded dimension. Theorem (R.Wagner '19)

1. The attening of the hypercube with S00 produces a chain complex. Its homology, denoted is a knot invariant which Hgl0 categories the Alexander polynomial. 2. There is a spectral sequence from the reduced triply graded homology to . Hgl0 Thank you !!