Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical...

1 / 47

Khovanov homology of framed and signed chord diagrams.

Oleg Viro

December 2, 2006

Knots and links

Knots and links• Classical linkdiagrams

• 1D-picture

• Gauss diagram

• Reconstruction ofknot

Virtual links

Moves

Kauffman bracket

Gauss diagrams of apoor man

Khovanov homology

Orientation of chorddiagrams

Khovanov complex offramed chord diagram

2 / 47

Classical link diagrams


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



3 / 47

A knot is a smooth simple closed curve in the 3-space.



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



3 / 47


That is a circle smoothly embedded into R3 .



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



3 / 47


A link is a union of several disjoint knots.



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



3 / 47



To describe a knot graphically, project it to a plane



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



3 / 47




and decorate at double points to show over- and

under-passes.



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



3 / 47





under-passes.



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



3 / 47





under-passes. This gives rise to a knot diagram:



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



3 / 47








• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



3 / 47






A link diagram:

1D-picture


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



4 / 47

A knot diagram

1D-picture


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



4 / 47

A knot diagram is a 2D picture of knot.

1D-picture


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



4 / 47


In many cases 1D picture serves better.

1D-picture


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



4 / 47



1D picture comes from a parameterization.

1D-picture


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



4 / 47



1D picture comes from a parameterization.

initial

point

12

3

4

1

23

4

2

1 4

3

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47

Decorate the source:

initial

point

12

3

4

1

23

4

2

1 4

3

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47


• with arrows from overpass to underpass,

initial

point

12

3

4

1

23

4

2

1 4

3

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47



initial

point

12

3

4

1

23

4

2

1 4

3

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47



• with the signs of crossings

initial

point

12

3

4

1

23

4

2

1 4

3

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47




1

23

4

2

1 4

3

+−−

+

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47




1

23

4

2

1 4

3

+−−

+

Signs:

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47




1

23

4

2

1 4

3

+−−

+

Signs: positive

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47




1

23

4

2

1 4

3

+−−

+

Signs: positive , negative .

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47




1

23

4

2

1 4

3

+−−

+

Signs: positive , negative . The result

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47




1

23

4

2

1 4

3

+−−

+

Signs: positive , negative . The result,

1

23

4

2

1 4

3

+−−

+

Gauss diagram


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



5 / 47




1

23

4

2

1 4

3

+−−

+

Signs: positive , negative . The result,

1

23

4

2

1 4

3

+−−

+, is called a Gauss diagram of the knot.

Reconstruction of knot


• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

Take any such diagram, say,

1

+ +

−32

and try to reconstruct the knot.



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

Start with crossings:



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

Start with crossings:

2

3

1



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

Connect them step by step:

2

3

1



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

Connect them step by step:

2

3

1



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

The next step does not work!

2

3

1



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

But let us continue!

2

3

1



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

Yet another obstruction!

2

3

1



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

We did it!

2

3

1



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

We did it! But what is the result?

2

3

1



• 1D-picture

• Gauss diagram


Virtual links

Moves

Kauffman bracket


Khovanov homology



6 / 47

1

+ +

−32

We did it! But what is the result?

2

3

1

The result is called a virtual knot diagram.

Virtual links

Knots and links

Virtual links

• Virtual knot diagrams

• Diagram on a surface

Moves

Kauffman bracket


Khovanov homology



7 / 47

Virtual knot diagrams

Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



8 / 47

A virtual knot diagram has crossings of 2 types:


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



8 / 47

A virtual knot diagram has crossings of 2 types:classical


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



8 / 47

A virtual knot diagram has crossings of 2 types:classical or real


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



8 / 47

A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagram


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



8 / 47

A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



8 / 47

A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



8 / 47

A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.Who can help to get rid of virtual crossings?


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



8 / 47

A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.Who can help to get rid of virtual crossings?Handles!


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



8 / 47

A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.Who can help to get rid of virtual crossings?Handles!

2

3

1

Diagram on a surface

Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47

A knot diagram drawn on orientable surface S


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47

A knot diagram drawn on orientable surface S ,instead of the plane


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47

A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47

A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47

A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47

A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surface


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47

A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47


Virtual knot diagrams emerge as projections to plane


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47


Virtual knot diagrams emerge as projections to plane of knotdiagrams on a surface.


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47


Virtual knot diagrams emerge as projections to plane of knotdiagrams on a surface.The surfaces is not unique:


Knots and links

Virtual links



Moves

Kauffman bracket


Khovanov homology



9 / 47


Virtual knot diagrams emerge as projections to plane of knotdiagrams on a surface.The surfaces is not unique: one can add handles.

Moves

Knots and links

Virtual links

Moves

• Moves• Moves of virtual linkdiagram

• Moves of Gaussdiagrams

• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory

• Isotopy problem

Kauffman bracket


Khovanov homology



10 / 47

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47

What happens to a link diagram, when the link moves?

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47

What happens to a link diagram, when the link moves?Link diagram moves, too.

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

(R1):

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

(R1):

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

(R1):

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

(R1):

(R2):

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

(R1):

(R2):

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

(R1):

(R2):

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

(R1):

(R2):

(R3):

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

(R1):

(R2):

(R3):

Moves

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



11 / 47


Reidemeister moves:

(R1):

(R2):

(R3):

Moves of virtual link diagram

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



12 / 47

A virtual link diagram

(i.e., a plane projection of a link diagram on a surface)

��

��

��

��


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



12 / 47

A virtual link diagram moves like this:

��

��

��

��


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



12 / 47


Reidemeister moves:

��

��

��

��


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



12 / 47


Reidemeister moves:

Virtual moves:

��

��

��

��


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



12 / 47


Reidemeister moves:

Virtual moves:

All virtual moves can be replaced by detour moves:

��

��

��

��

Moves of Gauss diagrams

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47

Gauss diagrams has nothing to do with virtual crossings!


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47

Gauss diagrams has nothing to do with virtual crossings!They do not change under virtual moves.


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47

Reidemeister moves acts on Gauss diagram:


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47


Move’sname

Reidemeistermove

Its action on Gauss diagram

Positivefirstmove

Nega-tive firstmove


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47


Move’sname

Reidemeistermove


Positivefirstmove

+

Nega-tive firstmove


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47


Move’sname

Reidemeistermove


Positivefirstmove

+

Nega-tive firstmove


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47


Move’sname

Reidemeistermove


Positivefirstmove

+

Nega-tive firstmove

−


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47


Move’sname

Reidemeistermove


Secondmove

Thirdmove


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47


Move’sname

Reidemeistermove


Secondmove

−ε

ε

Thirdmove


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47


Move’sname

Reidemeistermove


Secondmove

−ε

ε

Thirdmove


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



13 / 47


Move’sname

Reidemeistermove


Secondmove

−ε

ε

Thirdmove β

γ

α

β

γ

α

Combinatorial incarnation of knot theory

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



14 / 47


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



14 / 47

Classical Links → Link diagrams


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



14 / 47

Classical Links → Link diagramsIsotopies → Reidemeister moves


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



14 / 47


Combinatorial incarnations of virtual knot theory


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



14 / 47


Combinatorial incarnations of virtual knot theory

GaussDiagrams

←Virtual Links(?)

→Virtual LinkDiagrams

ReidemeisterMoves

←VirtualIstopies (?)

→Reidemeisterand Detourmoves

Topological meaning of virtual knot theory

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



15 / 47

Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



15 / 47


Virtual links up tovirtual isotopies

=

Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



15 / 47



=


Implies that virtual links generalize classical ones.


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



15 / 47



=



Bridges combinatorics


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



15 / 47



=



Bridges combinatorics (= 1D topology)


Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



15 / 47



=



Bridges combinatorics with (3D-) topology.

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47

Isotopy Problem:

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47

Isotopy Problem: Are given two classical links isotopic?

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47


Combinatorial reformulation:

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47


Combinatorial reformulation:Can given two Gauss diagrams be related by moves?

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47



Virtual Isotopy Problem:

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47



Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47




Invariants needed!

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47




Invariants needed!The most classical link invariant is the link group.

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47




Invariants needed!The most classical link invariant is the link group,the fundamental group of the link complement R

3r link .

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47




Invariants needed!The most classical link invariant is the link group.It was generalized.

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47




Invariants needed!The most classical link invariant is the link group.It was generalized, even in two ways!

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47




Invariants needed!The most classical link invariant is the link group.It was generalized: upper and lower!

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47




Invariants needed!The most classical link invariant is the link group.It was generalized: upper and lower!In terms of links in a thickened surface this is the fundamentalgroup of the complement, but with one of two sides of theboundary contracted to a point.

Isotopy problem

Knots and links

Virtual links

Moves




• Isotopy problem

Kauffman bracket


Khovanov homology



16 / 47




Invariants needed!The most classical link invariant is the link group.It was generalized: upper and lower!In terms of links in a thickened surface this is the fundamentalgroup of the complement, but with one of two sides of theboundary contracted to a point.

Kauffman bracket is more practical and elementary invariant.

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket

• Kauffman bracket

• Kauffman state sum. I• Kauffman state sum.II

• Example• Kauffman state summodel for Gaussdiagrams


Khovanov homology



17 / 47

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47

〈Link diagram〉 ∈ Z[A,A−1]

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


(a Laurent polynomial in A with integer coefficients).

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 =

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 =

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 =

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 =

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6

〈empty link〉 = 〈〉 =

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6

〈empty link〉 = 〈〉 = 1

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 =

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 =

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9

〈figure-eight knot〉 =

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9

〈figure-eight knot〉 = 〈〉 =

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9

〈figure-eight knot〉 = 〈〉 = − A10 − A−10

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9


Kauffman bracket is defined by the following properties:

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9


Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9


Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9


Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,3. 〈〉 = A〈〉+ A−1〈〉 (Kauffman Skein Relation) .

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9


Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,3. 〈〉 = A〈〉+ A−1〈〉 (Kauffman Skein Relation) .Uniqueness is obvious.

Kauffman bracket

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



18 / 47


〈unknot〉 = 〈©〉 = − A2 − A−2

〈Hopf link〉 = 〈〉 = A6 + A2 + A−2 + A−6


〈trefoil〉 = 〈〉 = A7 + A3 + A−1 − A−9


Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,3. 〈〉 = A〈〉+ A−1〈〉 (Kauffman Skein Relation) .Uniqueness is obvious.Invariant under R2 and R3, under R1 multiplies by −A±3.

Kauffman state sum. I

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



19 / 47

A state of diagram is a distribution of markers over allcrossings.


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



19 / 47


Knot diagram:


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



19 / 47


Knot diagram: and its states:


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



19 / 47



,


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



19 / 47



, ,


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



19 / 47



, , ,


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



19 / 47



, , , ,


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



19 / 47



, , , , . . .


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



19 / 47



, , , , . . .

Totally 2c states, where c is the number of crossings.

Kauffman state sum. II

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



20 / 47

Three numbers associated to a state s :


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



20 / 47


1. the number a(s) of positive markers ,


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



20 / 47



2. the number b(s) of negative markers ,


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



20 / 47




3. the number |s| of components of the curve obtained bysmoothing along the markers:


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



20 / 47





s = 7→


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



20 / 47





s = 7→ smoothing(s) =


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



20 / 47






|s| = 2


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



20 / 47






|s| = 2

State Sum: 〈D〉 =∑

s state of D Aa(s)−b(s)(−A2 − A−2)|s|

Example

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



21 / 47

Hopf link,

Example

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



21 / 47

Hopf link,⟨ ⟩

=

Example

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



21 / 47

Hopf link,⟨ ⟩

=

⟨ ⟩

+

⟨ ⟩

+

⟨ ⟩

+

⟨ ⟩

=

Example

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



21 / 47

Hopf link,⟨ ⟩

=

⟨ ⟩

+

⟨ ⟩

+

⟨ ⟩

+

⟨ ⟩

=

A2(−A2−A−2)2 + 2(−A2−A−2) + A−2(−A2−A−2)2 =

Example

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



21 / 47

Hopf link,⟨ ⟩

=

⟨ ⟩

+

⟨ ⟩

+

⟨ ⟩

+

⟨ ⟩

=

A2(−A2−A−2)2 + 2(−A2−A−2) + A−2(−A2−A−2)2 =

A6 + A2 + A−2 + A−6

Kauffman state sum model for Gauss diagrams

Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



22 / 47

Crossing 7→ arrow.

7→−

7→+


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



22 / 47


7→−

7→+

Smoothing of a crossing 7→ a surgery along the arrow.


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



22 / 47


7→−

7→+


+7→

−7→

positive marker, positive crossing negative marker, negative crossing


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



22 / 47


7→−

7→+


+7→

−7→


+7→

−7→

positive marker, negative crossingnegative marker, positive crossing


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



22 / 47


7→−

7→+


+7→

−7→


+7→

−7→


Smoothing depends only of the signs of marker and crossing.


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



22 / 47


7→−

7→+


+7→

−7→


+7→

−7→


Smoothing depends only of the signs of marker and crossing.No need in direction of the arrow!


Knots and links

Virtual links

Moves

Kauffman bracket





Khovanov homology



22 / 47


7→−

7→+


+7→

−7→


+7→

−7→


Smoothing depends only of the signs of marker and crossing.No need in direction of the arrow!Kauffman state sum is defined for signed chord diagrams.

Gauss diagrams of a poorman

Knots and links

Virtual links

Moves

Kauffman bracket


• Signed chorddiagrams

• State of signed chorddiagram

• Smoothing of asigned chord diagram

• Framing

• Framed chorddiagrams

• Signed to framed

• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening

• A link in orientablethickening of anon-orientable surface

Khovanov homology



23 / 47

Signed chord diagrams

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



24 / 47

A chord diagram (B, c1, . . . , cn)


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



24 / 47

A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), and

B


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



24 / 47

A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



24 / 47

A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)• in which B is oriented and


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



24 / 47

A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)• in which B is oriented and• each chord is equipped with a sign

−+

−−−

+ +


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



24 / 47

A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)• in which B is oriented and• each chord is equipped with a signis called a signed chord diagram.

−+

−−−

+ +

State of signed chord diagram

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



25 / 47

A state of the signed chord diagramis a distribution of another collection of signs over the set of allchords.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



25 / 47

A state of the signed chord diagramis a distribution of another collection of signs over the set of allchords.These are marker signs,


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



25 / 47

A state of the signed chord diagramis a distribution of another collection of signs over the set of allchords.These are marker signs, the original signs are structure signs.

Smoothing of a signed chord diagram

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



26 / 47

A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



26 / 47


+ +

(−)

(+)

(+) (−)

−

−


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



26 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(+) (−)

−

−


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



26 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(+) (−)

−

−


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



26 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(+) (−)

−

−


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



26 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(+) (−)

−

−

Morse modification at a chord depends on its signs.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



26 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(+) (−)

−

−

Morse modification at a chord depends on its signs.Denote by σ the product of the structure and the marker signs.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



26 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(+) (−)

−

−

Morse modification at a chord depends on its signs.Denote by σ the product of the structure and the marker signs.If σ = + , the Morse modification preserves the structureorientation.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



26 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(+) (−)

−

−

Morse modification at a chord depends on its signs.Denote by σ the product of the structure and the marker signs.If σ = + , the Morse modification preserves the structureorientation.If σ = − , the Morse modification destroys the orientation.

Framing

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



27 / 47

A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.

Framing

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



27 / 47


−

+

Framing

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



27 / 47


−

+

+

−

Framing

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



27 / 47


−

+

If the link is not oriented, specify the framing on the chordsgiving positive smoothing.

Framing

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



27 / 47


−

+


Framing

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



27 / 47


−

+


shorthand notation

Framed chord diagrams

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



28 / 47



Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



28 / 47



Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



28 / 47

A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framing


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



28 / 47

A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framing


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



28 / 47

A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framingis called a framed chord diagram.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



28 / 47


Kauffman bracket state sum is defined for a framed chorddiagram.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



28 / 47


−−

+

−

+−

+

Kauffman bracket state sum is defined for a framed chorddiagram.A state is a distribution of signs over the set of chords.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



28 / 47


−−

+

−

+−

+

Kauffman bracket state sum is defined for a framed chorddiagram.A state is a distribution of signs over the set of chords.The smoothing defined by a state is according to the famingalong the chords marked with + and the opposite oneotherwise.

Signed to framed

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



29 / 47

A signed chord diagram turns canonically to a framed one:

Signed to framed

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



29 / 47

A signed chord diagram turns canonically to a framed one:On a chord with + take the framing surgery along whichpreserves the orientation

Signed to framed

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



29 / 47

A signed chord diagram turns canonically to a framed one:On a chord with + take the framing surgery along whichpreserves the orientation,on a chord with − take the framing surgery along whichreverses the orientation.

Signed to framed

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



29 / 47

A signed chord diagram turns canonically to a framed one:On a chord with + take the framing surgery along whichpreserves the orientation,on a chord with − take the framing surgery along whichreverses the orientation.Forget the orientation.

Orientable thickenings of non-orientable surfaces

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



30 / 47

Non-orientable surface can be thickenedto an oriented 3-manifold!


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



30 / 47

Non-orientable surface can be thickenedto an oriented 3-manifold!Example:Thicken a Mobius band M in R

3 .


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



30 / 47


3 .


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



30 / 47


3 .

A neighborhood of M in R3 is orientable and fibers over M .

Abstract construction of an orientable thickening

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



31 / 47

Thicken a non-orientable surface S :


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



31 / 47

Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



31 / 47


C


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



31 / 47


C

2. Cut S along C : S 7→ S C


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



31 / 47


C


3. Thicken: (S C)× R .


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



31 / 47


C


3. Thicken: (S C)× R .4. Paste over the sides of the cut (x+, t) ∼ (x−,−t) .

A link in orientable thickening of a non-orientablesurface

Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



32 / 47

A diagram on the surface.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



32 / 47

A diagram on the surface.Reidemeister moves plus two more moves:

and


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



32 / 47


and

Twisted Gaussdiagram

=Gauss diagram with a finiteset of dots marked on thecircle.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



32 / 47


and



Two more moves:

and ε ε


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



32 / 47


and



Two more moves:

and ε ε

Forgetting dots and arrows turns a twisted Gauss diagram intoa signed chord diagram.


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



32 / 47


and



Two more moves:

and ε ε

Forgetting dots and arrows turns a twisted Gauss diagram intoa signed chord diagram.(together with moves)


Knots and links

Virtual links

Moves

Kauffman bracket





• Framing





Khovanov homology



32 / 47


and



Two more moves:

and ε ε

Forgetting dots and arrows turns a twisted Gauss diagram intoa signed chord diagram.Corollary (Bourgoin). Links in orientable thickenings ofsurfaces have well-defined Kauffman bracket.

Khovanov homology

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology

• Khovanov homology

• Enhanced states

• Khovanov complex• More algebraicconstruction• What about virtuallinks?



33 / 47

Khovanov homology

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




34 / 47

Khovanov homology categorifies Jones polynomial.

Khovanov homology

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




34 / 47

Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.

Khovanov homology

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




34 / 47


D 7→ Hp,q(D) , 〈D〉 =∑

p,q(−1)pAq rk Hp,q(D) .

Khovanov homology

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




34 / 47


D 7→ Hp,q(D) , 〈D〉 =∑

p,q(−1)pAq rk Hp,q(D) . Relationto the original Khovanov homology:

Khovanov homology

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




34 / 47


D 7→ Hp,q(D) , 〈D〉 =∑


Hp,q(D) = Hw(D)−q−2p

2,3w(D)−q

2 (D)

Khovanov homology

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




34 / 47


D 7→ Hp,q(D) , 〈D〉 =∑



2,3w(D)−q

2 (D) , or

Hi,j(D) = Hj−i−w(D),3w(D)−2j(D).

Khovanov homology

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




34 / 47


D 7→ Hp,q(D) , 〈D〉 =∑



2,3w(D)−q

2 (D) , or

Hi,j(D) = Hj−i−w(D),3w(D)−2j(D).

In other words: Hp,q(D) = Hi,j(D) iff

q + 2j = 3w(D) and j − i + p = w(D) .

Enhanced states

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




35 / 47

Enhance states involved in the Kauffman state sum byattaching a sign to each component of the smoothing alongthe state.

Enhanced states

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




35 / 47


For example: state with smoothing

gives rise to 4 enhanced states

Enhanced states

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




35 / 47


For example: state with smoothing

gives rise to 4 enhanced states

−

+

+

+

+

−

−

−

Khovanov complex

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




36 / 47

For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).

Khovanov complex

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




36 / 47


〈D〉 =∑

S enhanced state of D〈S〉

Khovanov complex

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




36 / 47


〈D〉 =∑

S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:

τ(S) = p and a(S)− b(S)− 2τ(S) = q .

Khovanov complex

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




36 / 47


〈D〉 =∑


τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =

∑

p,q(−1)pAq rk Cp,q(D) .

Khovanov complex

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




36 / 47


〈D〉 =∑



∑

p,q(−1)pAq rk Cp,q(D) .Any differential ∂ : Cp,q(D)→ Cp−1,q(D) gives homologyHp,q(D) with 〈D〉 =

∑


Khovanov complex

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




36 / 47


〈D〉 =∑



∑


∑


Invariance of Hp,q(D) under Reidemeister moves wanted!

Khovanov complex

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




36 / 47


〈D〉 =∑



∑


∑



∂(S) =∑

±T with T , which differ from S by a singlemarker and appropriate signs on the circles passing near thevertex.

Khovanov complex

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




36 / 47


〈D〉 =∑



∑


∑



∂(S) =∑

±T with T , which differ from S by a singlemarker and appropriate signs on the circles passing near thevertex.

(|T | − |S|) = 1 is needed to have τ(T ) = τ(S)− 1 .

More algebraic construction

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




37 / 47

Let A be an algebra over Z generated by 1 and X withX2 = 0 .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




37 / 47

Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




37 / 47

Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:

∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




37 / 47


∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D

associate a copy of A with each component of Ds .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




37 / 47



associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




37 / 47



associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .Equip Vs with the second grading equal to the first gradingshifted by a(s)− b(s)− |s| .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




37 / 47




Then⊕p,qCp,q(D) = ⊕sVs


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




37 / 47




Then⊕p,qCp,q(D) = ⊕sVs

Differentials are defined by the multiplication andco-multiplication in A .

What about virtual links?

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47

This works for classical links, but does not for virtual!


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47

This works for classical links, but does not for virtual!

For virtual links, it works with Z2 coefficients.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47

Over integers d2 6= 0!


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47


Consider virtual diagram of the unknot:


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47


Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47



− A4 − 1 −−−→ −A2 − A−2

y

y

A4 + 2 + A−4 −−−→ −1− A−4


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47



− A4 − 1 −−−→ −A2 − A−2

y

y

A4 + 2 + A−4 −−−→ −1− A−4

Differentials are obvious in all A -components but the onecorresponding to A0 .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47



− A4 − 10

−−−→ −A2 − A−2

y0

y

A4 + 2 + A−4 −−−→ −1− A−4



Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47



10

−−−→

y0

y

1⊗X + X ⊗ 1 −−−→ 2× 1



Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Enhanced states




38 / 47



10

−−−→

y0

y

1⊗X + X ⊗ 1 −−−→ 2× 1

This does not happen if the chord diagram is orientable!


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology


• Orientation of a chorddiagram

• Obstruction toorientability

• Orientation of asmoothened chorddiagram


39 / 47

Orientation of a chord diagram

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47

= orientations of chords and arcs


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47


such that the chain with integer coefficients


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47


such that the chain with integer coefficients∑

arcs +∑

2 chords


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47



arcs +∑

2 chords is a cycle.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47



arcs +∑


That is ∂ (∑

arcs +∑

2 chords ) = 0.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47



arcs +∑


That is ∂ (∑

arcs +∑

2 chords ) = 0.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47



arcs +∑


That is ∂ (∑

arcs +∑

2 chords ) = 0.

A chord diagram is called orientable if it admits an orientation.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47



arcs +∑


That is ∂ (∑

arcs +∑

2 chords ) = 0.


Orientability of chord diagram with connected base is

equivalent to the following condition known to K.-F.Gauss:


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47



arcs +∑


That is ∂ (∑

arcs +∑

2 chords ) = 0.




The number of endpoints of chords on each arc bounded be

endpoints of a chord is even.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






40 / 47



arcs +∑


That is ∂ (∑

arcs +∑

2 chords ) = 0.




The number of endpoints of chords on each arc bounded be

endpoints of a chord is even.

The simplest nonorientable chord diagram: ⊗ .

Obstruction to orientability

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47

Try to orient a chord diagram.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47



Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47



Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47



Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47



Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47



Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47


We have met an obstruction.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47



The obstruction to orientability of a chord diagram

(B, c1, . . . , cn) is an element of H1(B,∪ni=1∂ci; Z2) .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47





Dual class belongs to H0(B r ∪ni=1∂ci; Z2) .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






41 / 47





Dual class belongs to H0(B r ∪ni=1∂ci; Z2) .

Orient the complement of the 0-cycle realizing it,

to get vice-orientation of the chord diagram.

Orientation of a smoothened chord diagram

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






42 / 47

If a chord diagram


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






42 / 47

If a chord diagram

+ +

(−)

(+)

(+) (−)

−

−


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






42 / 47

If a chord diagram is oriented,

+ +

(−)

(+)

(+) (−)

−

−


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






42 / 47


+ +

(−)

(+)

(+) (−)

−

−

its orientation induces an orientation of each result of itssmoothing.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






42 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(−)

−

−

(+)

its orientation induces an orientation of each result of itssmoothing.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






42 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(−)

−

−

(+)

its orientation induces an orientation of each result of itssmoothing.Similarly, a vice-orientation of a signed chord diagram inducesa vice-orientation of each result of its smoothing.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






42 / 47


+ +

(−)

(+)

(+) (−)

−

−

+ +

(−)

(+)

(−)

−

−

(+)

its orientation induces an orientation of each result of itssmoothing.Similarly, a vice-orientation of a signed chord diagram inducesa vice-orientation of each result of its smoothing.

Theorem (Manturov, Viro) Definition of the Khovanov complexextended straightforwardly to an oriented framed chorddiagram gives a complex invariant under Reidemeister movespreserving the orientation.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology



• Structure used in theconstruction• Involution in theFrobenius algebra

• Space associated toa state

• Partial differential

43 / 47

Structure used in the construction

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47

1. Framed chord diagram.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47

1. Framed chord diagram.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47

1. Framed chord diagram.2. Vice-orientation of the chord diagram.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47

1. Framed chord diagram.2. Vice-orientation of the chord diagram.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47

1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47



Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47


The chain groups are the same as in the Khovanovconstruction:


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47



⊕p,qCp,q(D) = ⊕sVs


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47




algebraically (up to isomorphisms).


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47




algebraically (up to isomorphisms).The structure is needed for a collection of the isomorphisms


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47




algebraically (up to isomorphisms).The structure is needed for a collection of the isomorphismsneeded for construction of differentials.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






44 / 47




algebraically (up to isomorphisms).The structure is needed for a collection of the isomorphismsneeded for construction of differentials.Homology does not depend on the structure.

Involution in the Frobenius algebra

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






45 / 47

Remind that A is a Frobenius algebra generated by 1 and X

with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:

∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






45 / 47



∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






45 / 47



∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .Notice: conj(ab) = conj(a) conj(b) .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






45 / 47



∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .Notice: conj(ab) = conj(a) conj(b) .But ∆(conj(1)) = ∆(1) = X ⊗ 1 + 1⊗X

= −∆(X)⊗∆(1)−∆(1)⊗∆(X) .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






45 / 47



∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .Notice: conj(ab) = conj(a) conj(b) .But ∆(conj(1)) = ∆(1) = X ⊗ 1 + 1⊗X

= −∆(X)⊗∆(1)−∆(1)⊗∆(X) .and∆(conj(X)) = ∆(−X) = −X ⊗X = −∆(X)⊗∆(X) .

Space associated to a state

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






46 / 47

Given a state s of a framed chord diagram D .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






46 / 47

Given a state s of a framed chord diagram D .Orient each connected component of Ds .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






46 / 47

Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






46 / 47

Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






46 / 47

Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






46 / 47

Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






46 / 47

Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.The results corresponding to the different choices of themare related by isomorphisms:


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






46 / 47

Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.The results corresponding to the different choices of themare related by isomorphisms:Reversing of orientation of a component corresponds to conjin the corresponding copy of A .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






46 / 47

Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.The results corresponding to the different choices of themare related by isomorphisms:Reversing of orientation of a component corresponds to conjin the corresponding copy of A .Permutations of the components corresponds to thepermutation isomorphism of the tensor product multiplied bythe sign of the permutation.

Partial differential

Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47

Let s and t be adjacent states of a framed chord diagram D

which is equipped with a vice orientation and markers.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).• Other components (which are common for Ds and Dt ) areto be ordered coherently.


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).• Other components (which are common for Ds and Dt ) areto be ordered coherently.Orient the first components according to thevice orientation at c .


Knots and links

Virtual links

Moves

Kauffman bracket


Khovanov homology






47 / 47


which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).• Other components (which are common for Ds and Dt ) areto be ordered coherently.Orient the first components according to thevice orientation at c . In these representations of Vs and Vt ,define the map by multiplication or co-multipication.

Date post:	18-Jun-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical...

Documents