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Categorification of 1 (And of the Alexander Polynomial) 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 , ¡ Æ ¡ 1 a a¡ P(unknot) 1 or P(unknot) ¡ . Æ Æ q q 1 ¡ ¡ HOMFLY-PT polynomial à ! à ! à ! 1 1 aP a¡ P (q q¡ )P , ¡ Æ ¡ 1 a a¡ P(unknot) 1 or P(unknot) ¡ . Æ Æ q q 1 ¡ ¡ N slN -invariants a q Æ General N: ReshetikhinTuraev sl -invariants (U (sl )), Ï N q N N 2: Jones polynomial, Ï Æ N 1: Trivial invariant P1 (1 on every link), Ï Æ N 0: Alexander polynomial. Ï Æ 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.  de Rham cohomology de Rahm, '31 $ Categorication Philosophy Promote numerical invariants to (graded) vector space valued invariants.  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.  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 $ à ! à ! à ! 1 1 qP1 q¡ P1 (q q¡ )P1 ¡ Æ ¡ The Alexander polynomial ¢ gl0-homology R.Wagner, '19. Ï $ à ! à ! à ! 1 ¢ ¢ (q q¡ )¢ ¡ Æ ¡ gl1 invariant Link diagram 1 -lin. comb. of plane graphs Z[q,q¡ ] 1 2 q¡ q¡ ¡ 1 2 qÅ qÅ ¡ plane graph element of 1 N[q,q¡ ] 2 ¡ 1¢#V (¡)/ #V (¡)/2 ¡ q q¡ [2] . Å Æ 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¡ Æ ¡ Å ¡ 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¡ Æ ¡ Å ¡ gl1 invariant Example 3 q¡ 3 3 4 2 5 6 1 q¡ [2] 3q¡ [2] 3q¡ [2] q¡ Æ ¡ Å ¡ 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¡ Æ ¡ Å ¡ Planar (vinyl) graph graded vector space dimension [2]#V (¡)/2 graded linear map gl1-homology Braid closure diagram hypercube of plane graphs graphs (with shifts) { 2} { 1} { 2} Å ¡! ¡ ¡ { 2} { 1} Å Å gl1-homology Braid closure diagram hypercube of plane graphs graphs (with shifts) { 2} { 1} { 2} Å ¡! ¡ ¡ { 2} { 1} Å Å Planar (vinyl) graph graded vector space dimension [2]#V (¡)/2 graded linear map 3 3 4 2 5 6 q¡ [2] 3q¡ [2] 3q¡ [2] q¡ ¡ Å ¡ 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¡ ¡ Å ¡ k Y #{ in Ci } Xi i 1 ¿(d,c) ÆY Æ (Xi Xj ) ¡ Ci Cj Dot conguration d . M D(¡) Q. Æ d Multiplication ¹ on D(¡). Coloring c (C1,C2,..., Ck ) F Æ ¡ i Ci . Æ Vinyl graph vector space Vinyl graph ¡ index k. k Y #{ in Ci } Xi i 1 ¿(d,c) ÆY Æ (Xi Xj ) ¡ Ci Cj M D(¡) Q. Æ d Multiplication ¹ on D(¡). Coloring c (C1,C2,..., Ck ) F Æ ¡ i Ci . Æ Vinyl graph vector space Vinyl graph ¡ index k. Dot conguration d . k Y #{ in Ci } Xi i 1 ¿(d,c) ÆY Æ (Xi Xj ) ¡ Ci Cj Multiplication ¹ on D(¡). Coloring c (C1,C2,..., Ck ) F Æ ¡ i Ci . Æ Vinyl graph vector space Vinyl graph ¡ index k. Dot conguration d . M D(¡) Q. Æ d k Y #{ in Ci } Xi i 1 ¿(d,c) ÆY Æ (Xi Xj ) ¡ Ci Cj Coloring c (C1,C2,..., Ck ) F Æ ¡ i Ci . Æ Vinyl graph vector space Vinyl graph ¡ index k. Dot conguration d . M D(¡) Q. Æ d Multiplication ¹ on D(¡). k Y #{ in Ci } Xi i 1 ¿(d,c) ÆY Æ (Xi Xj ) ¡ Ci Cj Vinyl graph vector space Vinyl graph ¡ index k. Dot conguration d . M D(¡) Q. Æ d Multiplication ¹ on D(¡). Coloring c (C1,C2,..., Ck ) F Æ ¡ i Ci . Æ Vinyl graph vector space Vinyl graph ¡ index k. Dot conguration d . M D(¡) Q. Æ d Multiplication ¹ on D(¡). Coloring c (C1,C2,..., Ck ) F Æ ¡ i Ci . Æ k Y #{ in Ci } Xi 2 2 2 i 1 X1 X2 X3X4 ¿(d,c) ÆY 3 2 Æ (Xi Xj ) Æ (X1 X2) (X3 X4) (X1 X4)(X2 X3) ¡ ¡ ¡ ¡ ¡ Ci Cj Vinyl graph vector space Vinyl graph ¡ index k. Dot conguration d . M D(¡) Q. Æ d Multiplication ¹ on D(¡). Coloring c (C1,C2,..., Ck ) F Æ ¡ i Ci . Æ k Y #{ in Ci } Xi 2 2 2 i 1 X1 X2 X3X4 ¿(d,c) ÆY 3 ¡ 2 Æ (Xi Xj ) Æ (X1 X2) (X3 X4) (X1 X4)(X2 X3) ¡ ¡ ¡ ¡ ¡ Ci Cj Vinyl graph vector space Vinyl graph ¡ index k. Dot conguration d . M D(¡) Q. Æ d Multiplication ¹ on D(¡). Coloring c (C1,C2,..., Ck ) F Æ ¡ i Ci . Æ k Y #{ in Ci } Xi 2 2 2 i 1 X X1 X2 X3X4 ¿(d,c) ÆY ¿(d) ¿(d,c) 3 2 Æ (Xi Xj ) Æ c col(¡) Æ (X1 X2) (X3 X4) (X1 X4)(X2 X3) ¡ 2 ¡ ¡ ¡ ¡ Ci Cj Vinyl graph vector space Proposition (R.Wagner, '17 ) S For any dot conguration d, ¿(d) Q[X1,...,Xk ] k . 2 S1(¡) D(¡)/ker(¿ ¹( , )X 0). Æ ± ²7! Theorem (R.Wagner, '18 ) #V (¡)/2 For any vinyl graph ¡, dimq S1(¡) [2] . Æ 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 ). Å Å 2. Hopf link: the Poincaré polynomial is 1 q2(1 t). Å Å Examples 4 2 1. Trefoil: the Poincaré polynomial is 1 q¡ (t t ). Å Å 2. Hopf link: the Poincaré polynomial is 1 q2(1 t). Å Å Do I have time? Alexander polynomial Marked ( ) braid closure 1 -lin. comb. of marked plane graphs ? Z[q,q¡ ] 1 q¡ ¡ 1 qÅ ¡ Marked plane graph element of 1 N[q,q¡ ] ¡ complicated (comes from Uq(gl(1 1)) mod). j ¡ 2 2 2 1 2 q 1 q¡ [2] 3q¡ [2] 3q¡ 0 ¡ Å Æ ¡ Å 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 ¡ Å Æ ¡ Å S00(¡ ) S1(¡) at least k 1 at ? { k 1} ? ⊆ Æ h ¡ i ¡ Å induced by S1. Theorem (R.Wagner, '19) For any right-marked vinyl graph ¡?, dimq S00(¡?) is the expected graded dimension. gl0-homology Same hypercube with a dierent functor. Theorem (R.Wagner, '19) For any right-marked vinyl graph ¡?, dimq S00(¡?) is the expected graded dimension. gl0-homology Same hypercube with a dierent functor. S00(¡ ) S1(¡) at least k 1 at ? { k 1} ? ⊆ Æ h ¡ i ¡ Å induced by S1. gl0-homology Same hypercube with a dierent functor. S00(¡ ) S1(¡) at least k 1 at ? { k 1} ? ⊆ Æ h ¡ i ¡ Å induced by S1. Theorem (R.Wagner, '19) For any right-marked vinyl graph ¡?, dimq S00(¡?) 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 !!.
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