A Diagramless Link Homology
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1 University of Iowa October 1, 2008
2 Introduction Definitions Link homologies are typically dependent upon link diagrams. Here we will develop a homology theory that is defined for equivalence classes of links under isotopy in S 3. The homology is built from a link equivalence class rather than a particular diagram for that link. In addition to defining this diagramless link homology theory, we will give examples of how one does calculations. Some conjectures will also be given.
3 Goeritz Matrix & Signature Given a surface F, the Goeritz matrix G F of F is the n n matrix whose (i, j) entry is lk(a i, τa j ), where the a i are generators for H 1 (F, Q), lk denotes linking number, and τa j is the pushoff of 2α j. The signature of a surface F, denoted by sig(f ), is defined to be the signature of the Goeritz matrix G F of that surface.
4 Goeritz Matrix & Signature Given a surface F, the Goeritz matrix G F of F is the n n matrix whose (i, j) entry is lk(a i, τa j ), where the a i are generators for H 1 (F, Q), lk denotes linking number, and τa j is the pushoff of 2α j. The signature of a surface F, denoted by sig(f ), is defined to be the signature of the Goeritz matrix G F of that surface. oeritz.pdf a b a b ( τa τb sig = (1) - (1) = 0 ).
5 Crosscuts Definitions Given a surface F with boundary, a crosscut c F is a properly embedded arc in F, i.e. c = c F. In other words there exists an embedding f : [0, 1] F with f ({0, 1}) = f ([0, 1]) F acrosscut.pdf. Crosscuts will be labeled as active or inactive. Active crosscuts will be denoted by a green color, and inactive crosscuts by a red color.
6 The Cross-Dual Definitions For a surface F with crosscuts, the cross-dual of F, denoted F cd, is the surface obtained by replacing each neighborhood of each crosscut in F by the corresponding piece of surface explained below: Replace each neighborhood of an active crosscut c by a copy of itself with a 1 2-right-handed twist, with two parallel crosscuts c and c present. Whether the crosscuts are inactive or active no longer matters. c c c
7 The Cross-Dual Definitions For a surface F with crosscuts, the cross-dual of F, denoted F cd, is the surface obtained by replacing each neighborhood of each crosscut in F by the corresponding piece of surface explained below: Replace each neighborhood of an inactive crosscut by a copy of itself with a 1 2-left-handed twist, with two parallel crosscuts c and c present. Whether the crosscuts are inactive or active no longer matters. c c c
8 Definition: Graph of the cross-dual The graph of a cross-dual surface is the union of its boundary and its crosscuts. That is, for a cross-dual surface F cd, graph of F cd = F cd {crosscuts of F cd }
9 Planarity of the graph of the cross-dual A graph is planar in a manifold M if there is a 2-sphere embedded in M that contains the graph.
10 Definition: D k -surface A D k -surface is a compact surface F with crosscuts {c 1,..., c k } so that the crosscuts are ordered and oriented, the components of F {c 1,..., c k }, called facets, are allowed to be decorated by dots (which are not allowed move between facets), and the cross-dual F cd is orientable and has a planar graph.
11 Definition: D k -surface A D k -surface is a compact surface F with crosscuts {c 1,..., c k } so that the crosscuts are ordered and oriented, the components of F {c 1,..., c k }, called facets, are allowed to be decorated by dots (which are not allowed move between facets), and the cross-dual F cd is orientable and has a planar graph.
12 The 4 Gradings Definitions Notation: δ(f) = # dots on F Our homology will be tetra-graded. Given a D k -surface F, define I(F ) := sig(f ). This will be our homological grading. J(F ) := χ(f ) sig(f ) + 2δ. This will be our polynomial grading. K (F) := k. This is the number of crosscuts on F. L(F ) := sig(f ) + (# of active crosscuts on F ).
13 Before Defining the Let F i,j,k,l be the free module of isotopy classes of D k -surfaces in S 3 with I = i, J = j, K = k, and L = l. (Note: These are not are chain groups yet. We first need to quotient out by some relations.)
14 The Relations, R Let R be the submodule generated by the following relations: (S0)
15 The Relations, R Let R be the submodule generated by the following relations: (S0) (S1)
16 The Relations, R Let R be the submodule generated by the following relations: (S0) (S1) (F2)
17 The Relations, R Let R be the submodule generated by the following relations: (S0) (S1) (F2) (NC) Remark: /
18 The : C i,j,k,l Define: C i,j,k,l := F i,j,k,l /R.
19 The : C i,j,k,l acrosscut.pdf Define: C i,j,k,l := F i,j,k,l /R. This is an element in C i,j,k,l, with i = sig = 1, j = sig χ + 2δ = = 1, k = #crosscuts = 1, l = sig + # active crosscuts = = 1.
20 We now begin the process of defining our differential map d : C i,j,k,l C i+1,j,k,l. d will be defined in parts, acting locally on neighborhoods of active crosscuts on D k -surfaces in C i,j,k,l. Given a D k -surface F C i,j,k,l with an active (and oriented) crosscut c on F, we define d c to be the map which replaces a neighborhood of c in F with the piece of surface shown below: c d c
21 front / back issue Lemma: The map d c is well-defined. That is, one gets the same surface when applying d c to either side of a nieghborhood of the crosscut c. Proof : A front side c B about z axis back side c A B A front side B these are the same back side A B
22 The rest of well-defined We must show that d c is well-defined on C i,j,k,l C i+1,j,k,l. This includes proving:
23 The rest of well-defined We must show that d c is well-defined on C i,j,k,l C i+1,j,k,l. This includes proving: F C i,j,k,l d c (F ) C i+1,j,k,l
24 The rest of well-defined We must show that d c is well-defined on C i,j,k,l C i+1,j,k,l. This includes proving: F C i,j,k,l d c (F ) C i+1,j,k,l 1 F is a D k -surface d c (F ) is a D k -surface
25 The rest of well-defined We must show that d c is well-defined on C i,j,k,l C i+1,j,k,l. This includes proving: F C i,j,k,l d c (F ) C i+1,j,k,l 1 F is a D k -surface d c (F ) is a D k -surface 2 d c increases I by +1 and preserves J, K & L.
26 The rest of well-defined We must show that d c is well-defined on C i,j,k,l C i+1,j,k,l. This includes proving: F C i,j,k,l d c (F ) C i+1,j,k,l 1 F is a D k -surface d c (F ) is a D k -surface 2 d c increases I by +1 and preserves J, K & L. d c applied to each relation in R equals 0.
27 The rest of well-defined We must show that d c is well-defined on C i,j,k,l C i+1,j,k,l. This includes proving: F C i,j,k,l d c (F ) C i+1,j,k,l 1 F is a D k -surface d c (F ) is a D k -surface 2 d c increases I by +1 and preserves J, K & L. d c applied to each relation in R equals 0. (NC) relation preserves I, J, K & L.
28 Proving the rest of well-defined Claim: F C i,j,k,l d c (F) C i+1,j,k,l 1 F is a D k -surface d c (F ) is a D k -surface
29 Proving the rest of well-defined Claim: F C i,j,k,l d c (F) C i+1,j,k,l 1 F is a D k -surface d c (F ) is a D k -surface F c d c d c (F) cross-dual cross-dual F cd (d c (F)) cd
30 Proving the rest of well-defined Claim: F C i,j,k,l d c (F) C i+1,j,k,l 1 F is a D k -surface d c (F ) is a D k -surface 2 d c increases I by +1 and preserves J, K & L.
31 Proving the rest of well-defined Claim: F C i,j,k,l d c (F) C i+1,j,k,l 1 F is a D k -surface d c (F ) is a D k -surface 2 d c increases I by +1 and preserves J, K & L. F c d c d c (F) F β 1 α 1 d c (F ) α0
32 Proving the rest of well-defined Claim: d c applied to each relation in R equals 0. Proof :
33 Proving the rest of well-defined Claim: (NC) relation preserves I, J, K & L. Proof : cdisksig.pdf
34 Proving the rest of well-defined Claim: (NC) relation preserves I, J, K & L. Proof :
35 And now d itself... Now that we have shown d c is well-defined in all ways, we move to define the differential map d. Definition: d := active crosscuts c on ( surfaces F C i,j,k,l ) ( 1) σ(c) d c, where σ(c) = (# of inactive crosscuts on F ordered to be less c).
36 d 2 = 0 Definitions Claim: d d = 0-map
37 d 2 = 0 Definitions Claim: d d = 0-map Proof : c 1 d c c 2 d c (Separate locations!)
38 d 2 = 0 Definitions Claim: d d = 0-map Proof : c 1 d c c 2 d c (Separate locations!)...we have Homology!
39 Notation: 1 1 x + x 1
40 Homology for the Unknot with k=1
41 A less trivial example unknot2.pdf
42 The Hopf Link, first example
43 The Hopf Link, second example
44 Conjectures Definitions There is an injective map from Khovanov Homology into this Diagramless Homology.
45 Conjectures Definitions There is an injective map from Khovanov Homology into this Diagramless Homology. Cor: Khovanov Homology is invariant under the Reidemeister moves.
46 Conjectures Definitions There is an injective map from Khovanov Homology into this Diagramless Homology. Cor: Khovanov Homology is invariant under the Reidemeister moves. There are a finite number of elements in H k (H k = H i,j,k,l, with only k fixed)
47 Conjectures Definitions There is an injective map from Khovanov Homology into this Diagramless Homology. Cor: Khovanov Homology is invariant under the Reidemeister moves. There are a finite number of elements in H k (H k = H i,j,k,l, with only k fixed) There is a way to restrict this homology to only get copies of Khovanov Homology
48 References Definitions [BN] [Fr] [G,L] [Wu] Dror Bar-Natan, Khovonov s Homology for Tangles and Cobordisms, Geometry & Topology, 9-33, (2005) Charles Frohman, One-sided Incompressible Surfaces in Seifert Fibered Spaces, Topology and its Applications, 23 (1986) C.McA. Gordon and R.A. Litherland, On the Signature of a Link, Invent. Math., 47 (1978) Ying-Qing Wu, On Planarity of Graphs in 3-Manifolds, Commentarii Mathematici Helvetici, Vol 67 (1992)
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