Evaluation of sl N -foams. Workshop on Quantum Topology Lille

Size: px

Start display at page:

Download "Evaluation of sl N -foams. Workshop on Quantum Topology Lille"

Ethelbert Stafford
5 years ago
Views:

1 Evaluation of sl N -foams Louis-Hadrien Robert Emmanuel Wagner Workshop on Quantum Topology Lille

2 a b a a + b b a + b b + c c a a + b + c

3 a b a a + b b a + b b + c c a a + b + c Definition (R. Wagner, 17) ( 1) 1 i<j N θ+ ij (F,c) f P f (c(f )) F N = c ( 1) N i=1 iχ(f i (c))/2 1 i<j N (X i X j ) χ(f ij (c)) 2

4 Definition (auffman Bracket, Jones polynomial) = 1 L = [2] q L = q J(L) = ( 1) n q n+ 2n D

5 Definition (auffman Bracket, Jones polynomial) = 1 L = [2] q L = q J(L) = ( 1) n q n+ 2n D = q q + q 2

6 Definition (auffman Bracket, Jones polynomial) = 1 L = [2] q L = q J(L) = ( 1) n q n+ 2n D = q q J ( ) = q 6 + q 4 + q q 2

7 hovanov homology ( ) F F(saddle) F(saddle) ( ) F {+1} ( ) F {+1} F(saddle) F F(saddle) ( ) {+2}

8 hovanov homology ( ) F F(saddle) F(saddle) ( ) F {+1} F(saddle) F is a TQFT ( ) F : 1-mfd vect. sp. F {+1} cob. lin. map F F(saddle) ( ) {+2}

9 hovanov homology ( ) F F(saddle) F(saddle) ( ) F {+1} ( ) F {+1} F(saddle) F F(saddle) ( ) {+2} Shift the homological degree by n, the q-degree by n + 2n. Take the homology.

10 Proposition (Bar-Natan, 02) hovanov homology is strictly stronger than the Jones polynomial (source

11 Proposition (Bar-Natan, 02) hovanov homology is strictly stronger than the Jones polynomial (source Theorem (ronheimer Mrowka, 10) hovanov homology detects the unknot.

ca/notplot/notserver/) Theorem (ronheimer Mrowka, 10) hovanov homology detects the

12 Proposition (Bar-Natan, 02) hovanov homology is strictly stronger than the Jones polynomial (source Theorem (ronheimer Mrowka, 10) hovanov homology detects the unknot. Milnor conjecture (ronheimer Mrowka, 93, Rasmussen 04) The slice genus of the (p, q)-torus knot is equal to (p 1)(q 1) 2.

13 Links hovanov homology Seq. of graded Z-mod Jones polynomial (sl 2 -invariant) χ q Laurent pol. A recipe to deal with crossings An ad-hoc TQFT

14 Links sl N -homology Seq. of graded Z-mod sl N -invariant χ q Laurent pol. A recipe to deal with crossings Rickard complexes An ad-hoc TQFT evaluation of foams

15 The sl N -link invariant Proposition (Drinfel d) One can deform U(sl N ) into H := U q (sl N ) such that it becomes a quasi-triangular Hopf C(q)-algebra with non-trivial braiding. k l id k q V, id ( l q V ) D 1 D 2 f 1 f 2 D 1 D 2 f 1 f 2 k l k l braiding

16 The sl N -link invariant Proposition (Drinfel d) One can deform U(sl N ) into H := U q (sl N ) such that it becomes a quasi-triangular Hopf C(q)-algebra with non-trivial braiding. k l id k q V, id ( l q V ) D 1 D 2 f 1 f 2 D 1 D 2 f 1 f 2 k l k l braiding k k evaluation k k coevaluation l l k + l k + l k k k qv l qv k+l q V k+l q V k qv l qv

17 MOY calculus (Murakami Ohtsuki Yamada) Lusztig ( 94): m n m = ( 1) m k q k m m n n + k m n + k m k k=max(0,m n) k n m m n m = ( 1) m k q m k m n n + k m n + k m k k=max(0,m n) k n m

18 k = [ ] N k q m m + n m n [ ] N m = n q m i j k j + k i + j + k = i j k m + n i + j m m + n i + j + k n = [ ] m + n m q m + n m n + l n + k m 1 m m + 1 m 1 n + k m + l k k + [N m 1] q = 1 m m 1 m m 1 m m = j=max (0,m n) [ ] l k j q m j 1 m n + l j m n + l + j n + j m n m + l n m + l

19 Wish: F : Foam N Z[X 1,..., X N ] mod gr MOY-graph graded module foam graded module map

20 Wish: F : Foam N Z[X 1,..., X N ] mod gr MOY-graph graded module foam graded module map Universal Construction An evaluation (Maybe) a TQFT

21 Wish: F : Foam N Z[X 1,..., X N ] mod gr MOY-graph graded module foam graded module map Universal Construction An evaluation (Maybe) a TQFT Theorem (R. Wagner, 17) The evaluation defined on the first slide together with the Universal Construction yields an ad-hoc TQFT.

22 Universal Construction (Blanchet, Habbeger, Masbaum, Vogel) Given: {closed cobordisms} R Γ F(Γ) := F Γ Γ 1 GΓ 2 F(G): R F ( F(Γ1 ) F(Γ 2 ) FΓ 1 FGΓ 2 )

23 Universal Construction (Blanchet, Habbeger, Masbaum, Vogel) Given: {closed cobordisms} R Γ F(Γ) := F Γ [ G Γ1 Γ 2 ] F(G): R F / ( F(Γ1 ) F(Γ 2 ) i λ if i = 0 if i λ iτ(f i G) = 0 for all Γ G [ F Γ1 ] [ FG Γ 2 ] )

24 a a+b b a a+b b = b+c a+b+c c a+b+c a+b c a b a b a + b = b + c a + b + c c a + b + c c

25 n+k n m n+k m k m+l n+l m+l k m = j=max(0,m n) α T (k j,l k+j) j n+k m n+k m k n+l m+l k ( 1) α +(l k+j)(m j) α cβ 1 β 2 cγ α 1 γ 2 β 1,β 2 γ 1,γ 2 m j γ2 π γ1 n+k n n+k m+j n+k m+j n+k m k m+l π β1 n+l+j π β2 m+l k n+j m

26 h\i A t 1 A b 1 A t 1 A b 2 A t 2 A b 1 A t 2 A b 2 B1 B2 C L R X αtαb h\i A1 A2 B t 1 B b 1 B t 1 B b 2 B t 2 B b 1 B t 2 B b 2 C L R X A t 1 A b 1 A1 A t 1 A b 2 A2 A t 2 A b 1 B t 1 B b 1 A t 2 A b 2 B t 1 B b 2 B1 B t 2 B b 1 B2 B t 2 B b 2 C C L L R R X X Table 1. Colorings of the foam G (the three pictures on the left in each cell) and Fα (the three pictures on the right in each cell) for all values of h and i. The set X consists of all the colors which so not belong to any facets, hence looking at the last column or the last line of this table, we have the definition of the colorings of Fα and G. A green cell indicates that both χhi and (θ + hi, θ hi ) change, the yellow only (θ+ hi, θ hi ) Table 1. Colorings of the foam E (the three pictures on the left in each cell) and F jtjb (the three pictures on the right in each cell) for all values of h and i. Bhe set X consists of all the colors which so not belong to any facets, hence looking at the last column or the last line of this table, we have the definition of the colorings of F jtjb and E. A green cell indicates that both αtαb χhi and (θ+ hi, θ hi ) change, the yellow only (θ+ hi, θ hi ) h\i A t 1 A b 1 A t 2 A b 2 B1 B2 t 1 A +1 1 A b 1 A t 2 A b B B Table 2. χhi(g(g)) χhi(fα(c)) h\i A t 1 A b 1 A t 2 A b 2 B1 B2 C t 1 A 0 0 A b 1 A t 2 A b 2 ±1 0 ±1 B1 ±1 +1 B C ±1 +1 Table 3. θ + hi (G, g) θ+ hi (Fα, c) h\i A1 A2 B1 b B1 t B2 t B2 b A A B1 t B1 b 1 +1 B2 t B2 b 1 +1 jtjb χhi(e(e)) χhi(f αtαb h\i A1 A2 B1 t B1 b B2 b B2 t C A1 0 0 A B1 t B1 b +1 ±1 B2 t B2 b ±1 0 ±1 C +1 ±1 θ + hi (E, e) θ+ jtjb hi (F c), αtαb

27 Proposition The module associated with a MOY-graph with a symmetry axis is a Frobenius algebra.

28 Proposition The module associated with a MOY-graph with a symmetry axis is a Frobenius algebra. Proposition (R. Wagner, 17) The Frobenius algebra associated with a1 + a2 a1 + a2 + a3 N a k a 1 a2 a3 a4 a k 1 N a k a1 + a 2 a1 + a 2 + a 3 N a k is isomorphic to the T -equivariant cohomology ring of Flag(C a 1 C a 1+a 2 C a 1+ +a k 1 C N ).

29 k π λi (X ai +1,..., X ai+1 ) i=1 a1 π λ1 a 2 a3 a 4 π λ2 πλ3 πλ4 a k 1 a k N π λk 1 πλk

30 Corollary (R. Wagner, 17) The Littlewood Richardson coefficients are given by: c λ αβ =( 1) λ +N(N+1)/2 a b π λ π α π β = ( 1) N(N+1)/2+ λ N A B={X 1,...,X N } A =a, B =b N ( 1) B<A a α(a)a β (A)a λ(b) (X 1,..., X N ).

31 Thank you!

Categorifying quantum knot invariants

Categorifying quantum knot invariants Ben Webster U. of Oregon November 26, 2010 Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, 2010 1 / 26 This talk is online at http://pages.uoregon.edu/bwebster/rims-iii.pdf.