graphs on surfaces and khovanov homology › ~abhijit › talks › dkh_slides.pdf · abhijit...

Graphs on surfaces and Khovanov homology

Abhijit Champanerkar

University of South Alabama

Joint work with Ilya Kofman and Neal Stoltzfus

Knotting Mathematics and ArtUniversity of South Florida

Tampa, FLNov 1-4, 2007

Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 1 / 16

Outline

1 Ribbon graphs

2 Quasi-trees

3 Quasi-trees and spanning trees

4 Quasi-trees and chord diagram

5 Main Theorem & Corollaries

Ribbon graphs

An (oriented) ribbon graph G is a multi-graph (loops and multiple edgesallowed) that is embedded in an oriented surface, such that itscomplement is a union of 2-cells.

Example

Ribbon graphs

Example

Ribbon graphs

Example

Ribbon graphs

Example

Ribbon graphs

Algebraic definition

G can also be described by a triple of permutations (σ0, σ1, σ2) of the set{1, 2, . . . , 2n} such that

σ1 is a fixed point free involution.

σ0 ◦ σ1 ◦ σ2 = Identity

This triple gives a cell complex structure for the surface of G such that

Orbits of σ0 are vertices.

Orbits of σ1 are edges.

Orbits of σ2 are faces.

Ribbon graphs

Algebraic definition

G can also be described by a triple of permutations (σ0, σ1, σ2) of the set{1, 2, . . . , 2n} such that

σ1 is a fixed point free involution.

σ0 ◦ σ1 ◦ σ2 = Identity

This triple gives a cell complex structure for the surface of G such that

Orbits of σ0 are vertices.

Orbits of σ1 are edges.

Orbits of σ2 are faces.

Ribbon graphs

Example

Ribbon graphs

Example

Ribbon graphs

Example

σ0 = (1234)(56) σ0 = (1234)(56)σ1 = (14)(25)(36) σ1 = (13)(26)(45)σ2 = (246)(35) σ2 = (152364)

Quasi-trees

Spanning trees and regular neighbourhoods

A spanning tree of a graph is a spanning subgraph without any cycles.

For a planar graph, a spanning tree is a spanning subgraph whose regularneighbourhood has one boundary component.

ExampleAbhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 6 / 16

Quasi-trees

Example

Quasi-trees

Example

Quasi-trees

Example

Quasi-trees

Example

Quasi-trees

A quasi-tree of a ribbon graph is a spanning ribbon subgraph with oneface.

Example

Quasi-trees

Example

Quasi-trees

Example

Quasi-trees

Example

Quasi-trees and spanning trees

Links, Tait graphs and ribbon graphs

Let D be a connected link diagram, let G be its Tait graph and G be itsall-A ribbon graph.

Theorem (C-Kofman-Stoltzfus) Quasi-trees of G are in one-onecorrespondence with spanning trees of G :

Qj ↔ Tv where v + j =V (G ) + E+(G )− V (G)

Qj denotes a quasi-tree of genus j , and Tv denotes a spanning tree with vpositive edges.

Spanning subgraphs H ←→ Kauffman states of D

H ←→ s(e) = B iff e ∈ H

# faces of H = # components of s

Quasi-trees ←→ States with one component

←→ Jordan trails of D

←→ Spanning trees of G

H ←→ s(e) = B iff e ∈ H

Spanning trees and Khovanov homology

Theorem (C-Kofman’04) For a knot diagram D, there exists a spanningtree complex C (D) = {Cu

v (D), ∂} with ∂ : Cuv → Cu−1

v−1 that is a

deformation retract of the reduced Khovanov complex C̃ (D). In particular,

H̃ i ,j(D; Z) ∼= Huv (C (D); Z)

with the indices related as: u = j − i + C1 & v = j/2− i + C2

The u-grading for spanning trees is obtained from “activities” ofedges of the Tait graph.

Wehrli also proved a similar result.

Question Is there an intrinsic way to understand the u-grading forquasi-trees ?

v (D), ∂} with ∂ : Cuv → Cu−1

v−1 that is a

H̃ i ,j(D; Z) ∼= Huv (C (D); Z)

v (D), ∂} with ∂ : Cuv → Cu−1

v−1 that is a

H̃ i ,j(D; Z) ∼= Huv (C (D); Z)

Quasi-trees and chord diagram

Quasi-trees and chord diagrams

Proposition Every quasi-tree Q corresponds to the ordered chord diagramCQ with consecutive markings in the positive direction given by thepermutation:

σ(i) =

{σ0(i) i /∈ Qσ−1

2 (i) i ∈ Q

σ(i) =

{σ0(i) i /∈ Qσ−1

2 (i) i ∈ Q

σ(i) =

{σ0(i) i /∈ Qσ−1

2 (i) i ∈ Q

σ(i) =

{σ0(i) i /∈ Qσ−1

2 (i) i ∈ Q

σ(i) =

{σ0(i) i /∈ Qσ−1

2 (i) i ∈ Q

The bigrading on quasi-trees

A chord in CQ is live if it does not intersect any lower-ordered chords,otherwise it is dead.

An edge of a quasi-tree is live or dead if th corresponding chord chord islive or dead.

Definition For any quasi-tree Q of G, we define

u(Q) = #{live edges not in Q} −#{live edges in Q}, v(Q) = −g(Q)

Define C(G) = ⊕u,vCuv (G), where

Cuv (G) = Z〈Q ⊂ G| u(Q) = u, v(Q) = v〉

Main Theorem & Corollaries

Main Theorem

Theorem (C-Kofman-Stoltzfus) For a knot diagram D, there exists aquasi-tree complex C(G) = {Cu

v (G), ∂} with ∂ : Cuv → Cu−1

v−1 that is adeformation retract of the reduced Khovanov complex. In particular, thereduced Khovanov homology H̃ i ,j(D; Z) is given by

H̃ i ,j(D; Z) ∼= Huv (C(G); Z)

with the indices related as follows:

u = j − i − w(D) + 1 and v = j/2− i + (V (G)− c+(D))/2

Corollaries

For any link L, the Turaev genus gT (L) is the minimum value of the genusof the all-A ribbon graphs over all diagrams of L.

Corollary The thickness of the reduced Khovanov homology of K is lessthan or equal to gT (K ) + 1.

Corollary The thickness of the (unreduced) Khovanov homology of K isless than or equal to gT (K ) + 2.

Corollary The Turaev genus of (3, q)-torus links is unbounded.

Corollaries

Thank You Very Much

Domo Arigato

Hvala Puno

Barak Llahu Fik

Bolshoe Spasibo

Dziekuje

graphs on surfaces and khovanov homology › ~abhijit › talks › dkh_slides.pdf · abhijit...

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