![Page 1: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/1.jpg)
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
![Page 2: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/2.jpg)
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
![Page 3: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/3.jpg)
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
![Page 4: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/4.jpg)
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
![Page 5: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/5.jpg)
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
![Page 6: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/6.jpg)
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
![Page 7: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/7.jpg)
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
![Page 8: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/8.jpg)
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
![Page 9: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/9.jpg)
Ribbon graphs
Example
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 5 / 16
![Page 10: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/10.jpg)
Ribbon graphs
Example
43
125
6
6
1
3
4
2
5
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 5 / 16
![Page 11: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/11.jpg)
Ribbon graphs
Example
43
125
6
6
1
3
4
2
5
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
![Page 12: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/12.jpg)
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
![Page 13: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/13.jpg)
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
![Page 14: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/14.jpg)
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
![Page 15: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/15.jpg)
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
![Page 16: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/16.jpg)
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
![Page 17: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/17.jpg)
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
![Page 18: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/18.jpg)
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
![Page 19: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/19.jpg)
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
![Page 20: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/20.jpg)
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
![Page 21: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/21.jpg)
Quasi-trees and spanning trees
Links, Tait graphs and ribbon graphs
3b
21
4b
1
1 2
3
4
3
4
3
8
1 2
1
6
7
2
3
8
4
2 4
5
7
65
6 8
5 7
1 2
3 4
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 8 / 16
![Page 22: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/22.jpg)
Quasi-trees and spanning trees
Links, Tait graphs and ribbon graphs
3b
21
4b
1
1 2
3
4
3
4
3
8
1 2
1
6
7
2
3
8
4
2 4
5
7
65
6 8
5 7
1 2
3 4
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 8 / 16
![Page 23: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/23.jpg)
Quasi-trees and spanning trees
Links, Tait graphs and ribbon graphs
3b
21
4b
1
1 2
3
4
3
4
3
8
1 2
1
6
7
2
3
8
4
2 4
5
7
65
6 8
5 7
1 2
3 4
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 8 / 16
![Page 24: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/24.jpg)
Quasi-trees and spanning trees
Links, Tait graphs and ribbon graphs
3b
21
4b
1
1 2
3
4
3
4
3
8
1 2
1
6
7
2
3
8
4
2 4
5
7
65
6 8
5 7
1 2
3 4
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 8 / 16
![Page 25: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/25.jpg)
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
![Page 26: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/26.jpg)
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
![Page 27: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/27.jpg)
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
![Page 28: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/28.jpg)
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
![Page 29: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/29.jpg)
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
![Page 30: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/30.jpg)
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
![Page 31: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/31.jpg)
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
![Page 32: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/32.jpg)
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
![Page 33: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/33.jpg)
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
![Page 34: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/34.jpg)
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
4
7
1
2
3
6
8
5
1
6
7
2
5
4
3
8
1
2
7c
4c
8c
3c
5
6
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 12 / 16
![Page 35: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/35.jpg)
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
4
7
1
2
3
6
8
5
1
2
7c
4c
8c
3c
5
6
1
6
7
2
5
4
3
8
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 12 / 16
![Page 36: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/36.jpg)
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
4
7
1
2
3
6
8
5
1
6
7
2
5
4
3
8
1
2
7c
4c
8c
3c
5
6
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 12 / 16
![Page 37: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/37.jpg)
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
4
7
1
2
3
6
8
5
1
6
7
2
5
4
3
8
1
2
7c
4c
8c
3c
5
6
Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 12 / 16
![Page 38: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/38.jpg)
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
![Page 39: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/39.jpg)
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
![Page 40: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/40.jpg)
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
![Page 41: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/41.jpg)
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
![Page 42: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/42.jpg)
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
![Page 43: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/43.jpg)
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
![Page 44: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/44.jpg)
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
![Page 45: Graphs on surfaces and Khovanov homology › ~abhijit › talks › dkh_slides.pdf · Abhijit Champanerkar (USA) Graphs on surfaces and Khovanov homology 11 / 16. Quasi-trees and](https://reader033.vdocuments.site/reader033/viewer/2022060506/5f1f405f0fc71161287522bb/html5/thumbnails/45.jpg)
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