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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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 2 / 16
Ribbon graphs
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 3 / 16
Ribbon graphs
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 3 / 16
Ribbon graphs
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 3 / 16
Ribbon graphs
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 3 / 16
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.
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 4 / 16
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.
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 4 / 16
Ribbon graphs
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 5 / 16
Ribbon graphs
Example
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Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 5 / 16
Ribbon graphs
Example
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σ0 = (1234)(56) σ0 = (1234)(56)σ1 = (14)(25)(36) σ1 = (13)(26)(45)σ2 = (246)(35) σ2 = (152364)
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 5 / 16
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
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.
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 6 / 16
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.
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 6 / 16
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.
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 6 / 16
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.
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 6 / 16
Quasi-trees
Quasi-trees
A quasi-tree of a ribbon graph is a spanning ribbon subgraph with oneface.
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 7 / 16
Quasi-trees
Quasi-trees
A quasi-tree of a ribbon graph is a spanning ribbon subgraph with oneface.
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 7 / 16
Quasi-trees
Quasi-trees
A quasi-tree of a ribbon graph is a spanning ribbon subgraph with oneface.
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 7 / 16
Quasi-trees
Quasi-trees
A quasi-tree of a ribbon graph is a spanning ribbon subgraph with oneface.
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 7 / 16
Quasi-trees and spanning trees
Links, Tait graphs and ribbon graphs
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Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 8 / 16
Quasi-trees and spanning trees
Links, Tait graphs and ribbon graphs
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Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 8 / 16
Quasi-trees and spanning trees
Links, Tait graphs and ribbon graphs
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Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 8 / 16
Quasi-trees and spanning trees
Links, Tait graphs and ribbon graphs
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Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 8 / 16
Quasi-trees and spanning trees
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)
2
Qj denotes a quasi-tree of genus j , and Tv denotes a spanning tree with vpositive edges.
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 9 / 16
Quasi-trees and spanning trees
Proof
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 10 / 16
Quasi-trees and spanning trees
Proof
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 10 / 16
Quasi-trees and spanning trees
Proof
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 10 / 16
Quasi-trees and spanning trees
Proof
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 10 / 16
Quasi-trees and spanning trees
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 ?
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16
Quasi-trees and spanning trees
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 ?
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16
Quasi-trees and spanning trees
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 ?
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 12 / 16
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
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Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 12 / 16
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
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Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 12 / 16
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
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Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 12 / 16
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
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Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 12 / 16
Quasi-trees and chord diagram
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〉
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 13 / 16
Quasi-trees and chord diagram
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〉
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 13 / 16
Quasi-trees and chord diagram
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〉
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 13 / 16
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
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 14 / 16
Main Theorem & Corollaries
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.
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 15 / 16
Main Theorem & Corollaries
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.
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 15 / 16
Main Theorem & Corollaries
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.
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 15 / 16
Main Theorem & Corollaries
Thank You Very Much
Domo Arigato
Hvala Puno
Barak Llahu Fik
Bolshoe Spasibo
Dziekuje
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 16 / 16
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