# Questions tagged [khovanov-homology]

Khovanov homology, constructed by Mikhail Khovanov, is a categorification of the Jones polynomial.

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### Are the tangle functors based off Khovanov homology braided monoidal functors?

I was wondering if the tangle functors constructed in "A functor-valued invariant of tangles" https://arxiv.org/pdf/math/0103190.pdf "An invariant of tangle cobordisms via subquotients of arc rings" ...
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### Khovanov Homology in Macaulay2

Has anyone ever written code for computing Khovanov homology in Macaulay2 or other similar software? I know there are various excellent programs for computing Khovanov homology, but I'm currently ...
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### Background needed to understand modern research on knot homology theories

I am a student of mathematics, and have some background in Algebraic Topology (Hatcher, Bott-Tu, Milnor-Stasheff), Differential Geometry (Lee, Kobayashi-Nomizu), Riemannian Geometry (Do Carmo), ...
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### Khovanov homology definition using vector spaces, Z-modules, abelian groups?

When I read various papers on Khovanov homology, sometimes it is defined in terms of graded vector spaces, sometimes as graded $\mathbb{Z}$-modules. Is there a difference? E.g. can the vector field ...
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### Categorifying skein algebras?

We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov ...
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### Relation between different versions of Bar-Natan homology

In Bar-Natan's paper: Khovanov’s homology for tangles and cobordisms, he defined a deformation of Khovanov homology. Namely, for any $m\geq 0$, Bar-Natan's homology $BN^{m}(K)$ is obtained by ...
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### Khovanov homology and Crane-Yetter TQFT

Crane-Yetter(-Kauffman) have constructed 4-dimensional TQFT in such a way that Reshetikhin-Turaev theory lives on the boundary $\partial M$ of a 4-manifold $M$. Therefore, Crane-Yetter TQFT can be ...
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### Why is Khovanov homology considered a 'categorification'?

I understand that the Euler characteristic of Khovanov homology is the Jones polynomial. But in what sense does this give category theory structure to the Jones polynomial, i.e., what are the objects ...
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### Invariance of Khovanov homology under first Reidemester move

I am studying Khovanov homology from five lectures on Khovanov homology and I want to try to show Khovanov homology is invariant under first Reidemester move but I cannot understand how we can write ...
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### Understanding “Decategorified” symplectic Khovanov homology

In http://arxiv.org/abs/math/0405089 Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was ...
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### Unnormalized Kauffman homology of the unknot

Is the unnormalized Kauffman homology of the unknot known? The Poincare polynomial of HOMFLY homology of the unknot is known as $$\frac{1+at}{1-q}.$$ Is the Poincare polynomials of Kauffman homology ...
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### Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...
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### Khovanov-Rozansky homology and spectral sequences

In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology of a knot, and ...
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### Computations and applications of Khovanov's functor valued invariant of tangles

Soon after his famous paper "A categorification of the Jones polynomial", Khovanov introduced a "bordered" version. His theory assingns to every oriented even tangle a complex of (H_n,H_m)-bimodules, ...
A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$. \begin{align} P_{0,0}&=1\\ \text{for $n\geq 1$}...
I'm interested in Lee's modification of Khovanov homology, which I'll denote $\operatorname{Kh}_{\operatorname{Lee}}^\ast$. Below $L$ is a link in $\mathbb R^3$. The groups \$\operatorname{Kh}_{\...