Do Super Cats Make Odd Knots?

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1 Do Super Cats Make Odd Knots? Sean Clark MPIM Oberseminar November 5, 2015 Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

2 ODD KNOT INVARIANTS Knots WHAT IS A KNOT? (The unknot) (The Trefoil Knot) Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

3 ODD KNOT INVARIANTS Knots WHAT IS A KNOT? (The unknot) Knots = { S 1 R 3} /isotopy 2D projection (avoiding triple intersections) (The Trefoil Knot) Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

4 ODD KNOT INVARIANTS Knots WHAT IS A KNOT? (The unknot) Knots = { S 1 R 3} /isotopy 2D projection (avoiding triple intersections) Knots are isotopic iff projections equivalent under planar isotopy + Reidemeister moves Useful tool for distinguishing knots: invariants! (The Trefoil Knot) Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

5 ODD KNOT INVARIANTS Knot Invariants JONES POLYNOMIAL AND KHOVANOV HOMOLOGY Example (V. Jones, 1984) Given a knot (or link) diagram D, there is a Laurent polynomial J D = J D (q) that is an invariant of knots. D = has J D = q + q 1. D = has J D = q 9 q 7 + q 5 + 2q 3 + q 1. Thus the trefoil is not the unknot! Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

6 ODD KNOT INVARIANTS Knot Invariants JONES POLYNOMIAL AND KHOVANOV HOMOLOGY Example (V. Jones, 1984) Given a knot (or link) diagram D, there is a Laurent polynomial J D = J D (q) that is an invariant of knots. D = has J D = q + q 1. Example (Khovanov, 2000) For a knot diagram D, construct complex [D] of graded v.s./k, subject to rules similar to Jones polynomial: [ ] = 0 k[1] k[-1] 0 = q + q 1 }{{} hdeg=0 Khovanov Homology (KH) is the homology of this complex. The graded Euler characteristic of KH = Jones polynomial! Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

7 ODD KNOT INVARIANTS Knot Invariants REPRESENTATION THEORY Example (Reshetikhin-Turaev, late 1980 s) Knots can be encoded in a category TAN of tangles. Given a nice Hopf algebra H and module V, can find a functor from TAN to H-REP. This defines a operator invariant of the knot. Special Case: The quantum group U q (sl 2 ) is a nice enough Hopf algebra. This procedure with simple 2-dim module yields a map Q(q) Q(q). Evaluation at 1 is the Jones polynomial! Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

8 ODD KNOT INVARIANTS Knot Invariants CATEGORIFICATION Both examples are categorifications: (1-cat) KH χ F U-mod (0-cat) Jones Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

9 ODD KNOT INVARIANTS Knot Invariants CATEGORIFICATION Both examples are categorifications: (2-cat) U-mod W K (1-cat) KH χ F U-mod (0-cat) Jones Linked via categorified quantum groups (for all colored invariants) Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

10 ODD KNOT INVARIANTS Knot Invariants CATEGORIFICATION Both examples are categorifications: (2-cat) U-mod? W K (1-cat) KH χ F U-mod OKH χ? (0-cat) Jones Jones Linked via categorified quantum groups (for all colored invariants) Question: Can we find similar explanation for OKH? Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

11 ODD KNOT INVARIANTS Knot Invariants CATEGORIFICATION Both examples are categorifications: (2-cat) U-mod U s -mod W K K (1-cat) KH χ F U-mod OKH χ F U s -mod (0-cat) Jones Jones Linked via categorified quantum groups (for all colored invariants) Question: Can we find similar explanation for OKH? Conjecture: Yes, with quantum osp(1 2n) (Lie superalgebra) Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

12 WHAT IS U s Let U s = U q (osp(1 2)) = Q(q) E, F, K, K 1, J with rel ns KK 1 = 1, KEK 1 = q 2 E, KFK 1 = q 2 F, EF + FE = J 2 = 1 and J is central. JK K 1 q q 1 Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

13 WHAT IS U s Let U s = U q (osp(1 2)) = Q(q) E, F, K, K 1, J with rel ns KK 1 = 1, KEK 1 = q 2 E, KFK 1 = q 2 F, EF πfe = J 2 = 1 and J is central, π = ±1. JK K 1 πq q 1 Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

14 WHAT IS U s Let U s = U q (osp(1 2)) = Q(q) E, F, K, K 1, J with rel ns KK 1 = 1, KEK 1 = q 2 E, KFK 1 = q 2 F, EF πfe = J 2 = 1 and J is central, π = ±1. JK K 1 πq q 1 There are important module homomorphisms: 1. R : X Y = Y X (R matrix) for any X, Y; satisfies braid rel ns. 2. There is a simple 2-dim. module V. Q(q) ɛ V V δ Q(q), Q(q) ɛ V V δ Q(q) δ ɛ = q + πq 1 = πδ ɛ Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

15 KNOT DIAGRAMS TO MORPHISMS Translate a knot diagram D a map Q(q) Q(q) ( constant): Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

16 KNOT DIAGRAMS TO MORPHISMS Translate a knot diagram D a map Q(q) Q(q) ( constant): Cut diagram into simple pieces Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

17 KNOT DIAGRAMS TO MORPHISMS Translate a knot diagram D a map Q(q) Q(q) ( constant): δ 1 δ 1 R R 1 R 1 ɛ ɛ Cut diagram into simple pieces Translate each slice into a morphism 1 = 1 V = 1 = 1 V = π ±1 R = πδ = δ = π 1 ɛ = ɛ = Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

18 KNOT DIAGRAMS TO MORPHISMS Translate a knot diagram D a map Q(q) Q(q) ( constant): δ 1 δ 1 R R 1 R 1 ɛ ɛ Cut diagram into simple pieces Translate each slice into a morphism 1 = 1 V = 1 = 1 V = π ±1 R = πδ = δ = π 1 ɛ = ɛ = Compose and scale by (πq) writhe Then we get the Jones polynomial in the variable π 1 q! Example: = π 1 (q + πq 1 ) = π 1 q + ( π 1 q) 1 = Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

19 HIGHER RANK AND/OR COLORED INVARIANTS Theorem (C) Let K be a knot, V(λ) a f.d. irrep. of U ns = U q (so(1 + 2n)) or U s = U q (osp(1 2n)), and J s/ns K (q) the corresponding colored knot invariant. Then JK s (q) = 1 JK ns( 1q). Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

20 HIGHER RANK AND/OR COLORED INVARIANTS Theorem (C) Let K be a knot, V(λ) a f.d. irrep. of U ns = U q (so(1 + 2n)) or U s = U q (osp(1 2n)), and J s/ns K (q) the corresponding colored knot invariant. Then JK s (q) = 1 JK ns( 1q). Main idea in proof: Complex isomorphism ψ : Û ns = Û s with ψ(q) = 1q. ψ induces a nice functor Ψ on a rep category ΨX = 1 XΨ where X = cup/cap/crossing Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

21 HIGHER RANK AND/OR COLORED INVARIANTS Theorem (C) Let K be a knot, V(λ) a f.d. irrep. of U ns = U q (so(1 + 2n)) or U s = U q (osp(1 2n)), and J s/ns K (q) the corresponding colored knot invariant. Then JK s (q) = 1 JK ns( 1q). Main idea in proof: Complex isomorphism ψ : Û ns = Û s with ψ(q) = 1q. ψ induces a nice functor Ψ on a rep category ΨX = 1 XΨ where X = cup/cap/crossing Conclusion: U s does not give new invariants. But it may lead to new odd knot homologies! Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

22 CURRENT INTERESTS Construct an odd analogue of Webster s construction. (An answer for the Jones polynomial would be nice!)? U s -mod OKH U s -mod Jones Studying these quantum groups at roots of unity. Further study of other types of quantum superalgebras. Categorification of quantum superalgebras and reps. Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

23 THANKS FOR YOUR ATTENTION! Selected References: S.C., Quantum osp(1 2n) knot invariants are the same as quantum so(2n + 1) invariants, arxiv: A. Ellis and A. Lauda, An odd categorification of U q(sl 2 ), to appear in Quantum Topology, arxiv: M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), V. Mikhaylov and E. Witten, Branes and supergroups, arxiv: P. Ozsváth, J. Rasmussen, and Z. Szabó, Odd Khovanov homology, arxiv: B. Webster, Knot invariants and higher representation theory, to appear in Memoirs of the AMS, arxiv: Sean Clark Do Super Cats Make Odd Knots? November 5, / 10

Quantum supergroups and canonical bases

Quantum supergroups and canonical bases Sean Clark University of Virginia Dissertation Defense April 4, 2014 WHAT IS A QUANTUM GROUP? A quantum group is a deformed universal enveloping algebra. WHAT IS