Do Super Cats Make Odd Knots? Sean Clark MPIM Oberseminar November - PowerPoint PPT Presentation

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

O DD K NOT I NVARIANTS Knots W HAT IS A KNOT ? (The unknot) (The Trefoil Knot) Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 2 / 10

O DD K NOT I NVARIANTS Knots W HAT IS A KNOT ? � → R 3 � S 1 ֒ ◮ Knots = / isotopy ◮ 2D projection (avoiding triple intersections) (The unknot) (The Trefoil Knot) Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 2 / 10

O DD K NOT I NVARIANTS Knots W HAT IS A KNOT ? � → R 3 � S 1 ֒ ◮ Knots = / isotopy ◮ 2D projection (avoiding triple intersections) (The unknot) ◮ 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, 2015 2 / 10

O DD K NOT I NVARIANTS Knot Invariants J ONES P OLYNOMIAL AND K HOVANOV H OMOLOGY 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. has J D = q + q − 1 . D = has J D = − q − 9 − q − 7 + q − 5 + 2 q − 3 + q − 1 . D = Thus the trefoil is not the unknot! Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 3 / 10

O DD K NOT I NVARIANTS Knot Invariants J ONES P OLYNOMIAL AND K HOVANOV H OMOLOGY 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. has J D = q + q − 1 . D = Example (Khovanov, 2000) For a knot diagram D , construct complex [ D ] of graded v.s./ k , subject to rules similar to Jones polynomial: → 0 “=” q + q − 1 ] = 0 → k [ 1 ] ⊕ k [ -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, 2015 3 / 10

O DD K NOT I NVARIANTS Knot Invariants R EPRESENTATION THEORY Example (Reshetikhin-Turaev, late 1980’s) Knots can be encoded in a category T AN of tangles . Given a “nice” Hopf algebra H and module V , can find a functor from T AN 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, 2015 4 / 10

O DD K NOT I NVARIANTS Knot Invariants C ATEGORIFICATION Both examples are categorifications : (1-cat) KH U -mod χ F (0-cat) Jones Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 5 / 10

O DD K NOT I NVARIANTS Knot Invariants C ATEGORIFICATION Both examples are categorifications : ˙ (2-cat) U -mod W K (1-cat) KH U -mod χ F (0-cat) Jones ◮ Linked via categorified quantum groups (for all colored invariants) Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 5 / 10

O DD K NOT I NVARIANTS Knot Invariants C ATEGORIFICATION Both examples are categorifications : ˙ (2-cat) U -mod ? W K (1-cat) KH U -mod OKH ? χ χ F (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, 2015 5 / 10

O DD K NOT I NVARIANTS Knot Invariants C ATEGORIFICATION Both examples are categorifications : ˙ ˙ (2-cat) U s -mod U -mod W K K U s -mod (1-cat) KH U -mod OKH χ χ F F (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 | 2 n ) (Lie super algebra) Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 5 / 10

Quantum osp ( 1 | 2 ) Super Categories W HAT 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 = JK − K − 1 − q − q − 1 J 2 = 1 and J is central. Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 6 / 10

Quantum osp ( 1 | 2 ) Super Categories W HAT 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 − π FE = JK − K − 1 π q − q − 1 J 2 = 1 and J is central, π = ± 1 . Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 6 / 10

Quantum osp ( 1 | 2 ) Super Categories W HAT 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 − π FE = JK − K − 1 π q − q − 1 J 2 = 1 and J is central, π = ± 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 ) ǫ ′ → V ⊗ V ∗ δ ′ → V ∗ ⊗ V ǫ δ Q ( q ) → Q ( q ) , → Q ( q ) δ ◦ ǫ = q + π q − 1 = πδ ′ ◦ ǫ ′ Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 6 / 10

Quantum osp ( 1 | 2 ) Super Categories K NOT D IAGRAMS TO M ORPHISMS Translate a knot diagram D � a map Q ( q ) → Q ( q ) ( constant): Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 7 / 10

Quantum osp ( 1 | 2 ) Super Categories K NOT D IAGRAMS TO M ORPHISMS 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, 2015 7 / 10

Quantum osp ( 1 | 2 ) Super Categories K NOT D IAGRAMS TO M ORPHISMS Translate a knot diagram D � a map Q ( q ) → Q ( q ) ( constant): ◮ Cut diagram into simple pieces ◮ Translate each slice into a morphism δ 1 ∗ = 1 V ∗ = 1 = 1 V = 1 ∗ ⊗ δ ⊗ 1 √ π ± 1 R = R ⊗ R √ πδ ′ = δ = 1 ∗ ⊗ R ⊗ 1 √ π − 1 ǫ = ǫ ′ = ǫ ⊗ ǫ Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 7 / 10

Quantum osp ( 1 | 2 ) Super Categories K NOT D IAGRAMS TO M ORPHISMS Translate a knot diagram D � a map Q ( q ) → Q ( q ) ( constant): ◮ Cut diagram into simple pieces ◮ Translate each slice into a morphism δ 1 ∗ = 1 V ∗ = 1 = 1 V = ◦ 1 ∗ ⊗ δ ⊗ 1 √ π ± 1 R = ◦ R ⊗ R √ πδ ′ = ◦ δ = 1 ∗ ⊗ R ⊗ 1 √ π − 1 ǫ = ǫ ′ = ◦ ǫ ⊗ ǫ ◮ Compose and scale by ( π q ) writhe Then we get the Jones polynomial in the variable √ π − 1 q ! = √ π − 1 ( q + π q − 1 ) = √ π − 1 q + ( √ π − 1 q ) − 1 = Example: Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 7 / 10

Quantum osp ( 1 | 2 ) Super Categories H IGHER 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 + 2 n )) or U s = U q ( osp ( 1 | 2 n )) , and J s / ns ( q ) the corresponding colored knot invariant. √ √ K ∗ J ns Then J s K ( q ) = − 1 K ( − 1 q ) . Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 8 / 10

Quantum osp ( 1 | 2 ) Super Categories H IGHER 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 + 2 n )) or U s = U q ( osp ( 1 | 2 n )) , and J s / ns ( q ) the corresponding colored knot invariant. √ √ K ∗ J ns Then J s K ( q ) = − 1 K ( − 1 q ) . Main idea in proof: √ U ns ∼ U s with ψ ( q ) = ◮ ∃ Complex isomorphism ψ : � = � − 1 q . ◮ ψ induces a nice functor Ψ on a rep category √ ∗ X Ψ where X = cup/cap/crossing ◮ Ψ X = − 1 Sean Clark Do Super Cats Make Odd Knots? November 5, 2015 8 / 10

Quantum osp ( 1 | 2 ) Super Categories H IGHER 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 + 2 n )) or U s = U q ( osp ( 1 | 2 n )) , and J s / ns ( q ) the corresponding colored knot invariant. √ √ K ∗ J ns Then J s K ( q ) = − 1 K ( − 1 q ) . Main idea in proof: √ U ns ∼ U s with ψ ( q ) = ◮ ∃ Complex isomorphism ψ : � = � − 1 q . ◮ ψ induces a nice functor Ψ on a rep category √ ∗ X Ψ where X = cup/cap/crossing ◮ Ψ X = − 1 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, 2015 8 / 10

Quantum osp ( 1 | 2 ) Super Categories C URRENT I NTERESTS ◮ Construct an odd analogue of Webster’s construction. (An answer for the Jones polynomial would be nice!) ˙ U s -mod ∃ ? U s -mod OKH ∃ ∃ 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, 2015 9 / 10