10 votes

Why is Khovanov homology considered a 'categorification'?

The idea is that Khovanov homology is a functor out of a category of tangle cobordisms. Bar-Natan's notes are a great reference for this. Khovanov (2002) sets this up in a "tangle 2-category:" the ...
  • 6,626
9 votes

Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?

Lisa Piccirillo just posted a few days ago this paper on the arXiv. Corollary 1.3 asserts exactly that $s$ is not a 0-trace invariant of the knot.
  • 9,439
8 votes

Understanding "Decategorified" symplectic Khovanov homology

One thing you didn't mention is the manifold in which this calculation happens: the Slodowy slice to a nilpotent of type $(n,n)$. This manifold is the key to everything, since it is a geometric avatar ...
  • 42.7k
7 votes

Questions on poincare homology spheres and branched covers

As Ian Agol points out the Orbifold Theorem will answer question 1 affirmatively. Although the original result is due to Thurston, the common references in the literature are: Boileau, Michel, ...
  • 5,171
5 votes

Version of Khovanov homology that does not produce torsion?

Odd Khovanov homology (arXiv.0710.4300) by Ozsvath, Rasmussen, and Szabo is a version of Khovanov homology that typically (but not always) has less torsion than the standard Khovanov homology. For ...
4 votes

Khovanov homology definition using vector spaces, Z-modules, abelian groups?

Yes, there is a difference- Khovanov homology over $\mathbb{Z}$ often has torsion elements of different orders which cannot all be visible over $\mathbb{F}_{p^k}$ for any $p$. See Shumakovitch https://...
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4 votes

Background needed to understand modern research on knot homology theories

I'm not an expert in Floer homology or Khovanov homology, but if that's your goal I don't think you need quite as wide a background as suggested in the other current answer (though admittedly that ...
3 votes

Khovanov Homology in Macaulay2

Adam, I'm not sure if you still need this but Sage has a program to compute the Khovanov homology of a link and has an interface with Macaulay 2/uses Macaulay 2. As a disclaimer, I am not an expert on ...
3 votes

Corollary in Rasmussen's paper about $s$-grading of Lee's canonical generators

Let $ C ( D) $ denote the Lee chain complex and $ Kh ' ( K ) $ its homology, $ q $ denote the grading on $ C ( D) $ (associated to the filtration) and $ s $ the induced grading on $ Kh ' ( K ) $. ...
3 votes

Invariance of Khovanov homology under first Reidemester move

I do not understand your example of Hopf link, since one cannot remove a crossing by the first Reidemeister move on this diagram. As a simple exercise to check the result, you can write down the chain ...
2 votes

Categorifying skein algebras?

Much of the research in knot homology has been about categorifying these algebras! Khovanov's paper math/0103190 is devoted to defining and studying the Temperley-Lieb 2-category which is a ...
1 vote

Khovanov-Rozansky/HOMFLY-PT homology in type B

Maybe take a look at the following paper of Rose and Tubbenhauer https://arxiv.org/abs/1908.06878.
1 vote

Background needed to understand modern research on knot homology theories

There have been several questions previously in this vein, but yours is more general. My present answer is adapted from an answer to a question asking for a "Road Map" to Homotopy Theory. Your ...
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