Knot homologies & lowdimensional. October 8, 2008 Salt Lake City, UT

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1 Knot homologies & lowdimensional topology ctober 8, 2008 Salt Lake City, UT

2 What is a knot? A knot is an equivalence class of smooth imbeddings f : S 1 R 3, where two imbeddings are equivalent if they can be connected by an isotopy.

3 Q: Why do we care about knots? Q: Why do I, Eli Grigsby, a lowdimensional topologist, care about knots? A: Fundamental for understanding 3- and 4-dimensional manifolds.

4 Q: Why 3- & 4-dimensional manifolds? Dimensions <3: Very constrained-- You Are Here too easy Dimensions 3, 4: Just right. Dimensions >4: Too much freedom-- translate questions to homotopy theory

5 Q: What do knots have to do with low-dimensional topology? A: They can be used to construct new 3-manifolds from old.

6 Surgery on Knots K R 3 R 3 N(K)

7 Surgery on Knots R 3 N(K) S 1 D 2

8 Surgery on Knots µ D 2 R 3 N(K) S 1 D 2 R 3 = [R 3 N(K)] µ=d 2 [S 1 D 2 ]

9 Surgery on Knots γ D 2 R 3 N(K) S 1 D 2 R 3 γ(k) = [R 3 N(K)] γ=d 2 [S 1 D 2 ] New 3-manifold!

10 Theorem (Lickorish-Wallace): Every closed, connected, oriented 3-manifold can be obtained by doing surgery on a link in S 3. Moral: If we can understand knots (links), we can understand 3-manifolds.

11 How do we understand knots? = Complicated...

12 Idea: Knot Invariants Algebraic bject Number Polynomial Group Graded group Category etc. Prove invariance under choice of diagram: Knot invariant

13 Classical knot invariants Alexander polynomial Jones polynomial Determinant Signature Arf invariant Mod p colorings Tristam-Levine signatures etc.

14 New knot invariants! Knot homology theories Powerful (separate knots, encode topological information) Computable (a computer can do it)

15 Knot homology theories H(C i,j, ) Homology of a Bi-graded chain complex

16 Knot homology theories H(C i,j, ) Knot Floer homology (P. zsváth-z. Szabó, Rasmussen) Khovanov homology (M. Khovanov) ĤF K i,j (K) Kh i,j (K)

17 Bi-graded chain complexes and Homology C i,j C i+1,j C i,j 1 C i+1,j C i,j is a finite-dimensional vector space (over F = Z/2Z) : C i,j C i,j 1 is a linear transformation = 0 Im() Ker() H i,j = Ker() Im()

18 Euler Characteristics: Knot homologies as Categorifications ĤF K i,j (K) i F K = F F F 2 F F F 2 F 4 j Alexander polynomial! : T 2 + T T + T 2

19 Knot Floer Homology categorifies the classical Alexander polynomial. Khovanov Homology categorifies the classical Jones polynomial.

20 Khovanov Homology & Knot Floer Homology Inspired by ideas in physics Categorify classical knot invariants Share other formal similarities Definitions are quite different...

21 Definition: Knot Floer Homology (Manolescu-zsváth-Sarkar)

22 Definition: Knot Floer Homology (Manolescu-zsváth-Sarkar)

23 Definition: Knot Floer Homology (Manolescu-zsváth-Sarkar)

24 Definition: Knot Floer Homology (Manolescu-zsváth-Sarkar) Grid Diagram for K

25 Grid Diagrams to Chain Complexes Generators: 1-1 matchings of red/blue lines Elements of S n Bijections of n-element set

26 Grid Diagrams to Chain Complexes Differentials: Count empty rectangles between generators differing by a transposition

27 Grid Diagrams to Chain Complexes Not empty: Contains an

28 Grid Diagrams to Chain Complexes Not empty: Contains an intersection point

29 Grid Diagrams to Chain Complexes Bi-grading: More complicated (but not hard)

30 Grid Diagrams to Chain Complexes Claim 1: 2 = 0. Not too hard. Claim 2: ĤF K(K) is independent of choice of grid diagram. Hard. H(C i,j, ) = ĤF K(K) V n 1

31 Definition: Khovanov Homology res. 3 1 res.

32 Definition: Khovanov Homology res. 3 1 res. (1,1,1) resolution

33 Definition: Khovanov Homology V k k circles V = Span F (v +, v )

34 Cube of Resolutions V V 2 V 2 V V 2 V V V 2

35 Cube of Resolutions V V 2 V 2 V V 2 V 3 = 0 V V 2

36 Cube of Resolutions V V 2 V 2 V V 2 V 3 = 0 V V 2 Diagramindendent Kh(K) := H(C i,j, ) homology

37 ne (of many) Applications Theorem (zsváth-szabó): ĤF K(K) detects the unknot. Conjecture: Kh(K) detects the unknot.

38 ne (of many) Applications Theorem (zsváth-szabó): ĤF K(K) detects the unknot. Theorem (G-, Wehrli): Kh n (K) detects the unknot (n 2).

39 Connections? Kh(K) and Double-branched cover (zsváth-szabó) ĤF (Y ) : 3-manifold invariant, related to knot invariant Kh(K) is related to ĤF (Σ(K)) K Doublebranched cover of K

40 Connections? Kh(K) and Double-branched cover V V 2 Kh(K) V 2 V V 2 V 3 V V 2

41 Connections? Kh(K) and Double-branched cover V V 2 ĤF (Σ(K)) V 2 V V 2 V 3 V V 2 Perturb the differential

42 Connections? Triply-graded superhomology? (Dunfield, Gukov, Rasmussen) Inspired by conjectured deep relationship in physics. H i,j,k (K) ĤF K(K) Kh(K) New Invariants ther Known Invariants

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