Twisting in Khovanov and Knot Floer Homology

Thomas Jaeger

Syracuse University

Monday, March 12, 2012

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Outline

Knot Floer Homology

Khovanov Homology

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Knot Floer Homology

Manolescu’s exact triangle

Goal: Find a link (in form of a spectral sequence) between Khovanov andKnot Floer Homology

Idea: Iterate Manolescu’s (δ-graded) skein exact triangle:

HFK ( )→ HFK ( )→ HFK ( )

This should give a spectral sequence whose E1-page is related to theKhovanov chain complex.

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Knot Floer Homology

Twisted Homology

One may hope, that the E2 page is related to Khovanov homology in asimilar manner.Unfortunately, the E2 page is not even a link invariant.

Baldwin and Levine’s idea is to consider twisted coefficients instead, whichkeeps track of (using an additional variable T how many timesholomorphic disks intersect points on the surface:

By judiciously placing those points, they can arrange that after tensoring

with k[T−1,T ]], all complete resolutions of the diagram have trivial HFK .

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Knot Floer Homology

Levine and Baldwin’s Spanning Tree Model

The resulting spectral sequence collapses at the E3 page for gradingreasons. The E1 = E2 page is the direct sum of Λ∗(VI ), where VI is avector space of dimension n − 1 associated to a connected resolutionindexed by I and n the number of crossings of the diagram. d2 can becomputed explicitly. It’s exact form is complicated, but it only relates aconnected resolution to its double successor:

This is called a spanning tree model since spanning trees are in(non-canonical) bijective correspondence with spanning trees of the black(or white) graph.

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Khovanov Homology

Khovanov HomologyWe wish to do something similar for Khovanov homology. For HFK , we

had to work hard to get a “cube-of-resolutions“ spectral sequence. InKhovanov homology, this comes for free since Khovanov homology isalready defined as a cube of resolutions: The (reduced) Khovanovhomology of a complete resolution with k components is

k[x1, x2, . . . , xk ]/(x1, x22 , x

23 , . . . , x

2k ).

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Khovanov Homology

Roberts’ Twisted Khovanov Homology

Notice that if we endow each resolution with a chain complex structurewhose differential is the endomorphism

∑i wixi , then this chain complex is

acyclic if at least one of the wi is invertible.

Place markings on the link diagram and assign variables x1, x2, . . . , xm andweights w1,w2, . . .wm to each of the markings. k[x1, · · · , xm] acts on theKhovanov complex and commutes with its differential.

We consider the double complex whose horizontal differential is the regularKhovanov differential and whose vertical differential is the endomorphism∑

wixi .

We want to arrange things so that the vertical homology of alldisconnected resolution is trivial. This can be done by ensuring that thereis a marking on each edge of the diagram, working over the ring k[t−1, t]]and setting wi = t i .

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Khovanov Homology

Example: Trefoil

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Khovanov Homology

A spanning tree modelThis double complex gives rise to a spectral sequence computing twisted

Khovanov homology. As for HFK , it converges at the E3 page for gradingreasons:

Its d2 differential can be computed explicitly using “Gaussian Elimination”:

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Khovanov Homology

What is twisted Khovanov homology?

Theorem (J.)

For knots, twisted Khovanov homology is isomorphic to (the original)Khovanov homology.

Proof.Move markings next to the basepoint:

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Khovanov Homology

Comparison with previous spanning tree models

As we have seen, twisted spanning tree models have differentials that canbe computed readily. This is in contrast with previous spanning treemodels by Ozsvath and Szabo for HFK and Champanerkar–Kofman andindependently Wehrli for Kh.

The downside is that we lose one grading: The twisted spanning treemodels compute only the δ-graded versions of the respective theories (theδ grading is characterized by the property that the action of the “edgering” on the chain complex has the same degree as the differential).

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012

Khovanov Homology

Comparison between HFK and Kh

I Given the obvious similiarities between the theories, one may wonderif they can be used to establish a spectral sequence relating them.Unfortunately,

I HFK computes not Knot Floer homology, but Knot Floer homologytensored with an exterior algebra over a vector space.

I Weights in HFK behave “multiplicatively” whereas weights in Khbehave “additively”.

I Both theories (conjecturally in case of HFK are invariant undermutation). Is there a way to see this using the spanning tree model?

Thomas Jaeger (Syracuse University) Twisting in Khovanov and Knot Floer Homology Monday, March 12, 2012