# Questions tagged [khovanov-homology]

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

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

In Jacob Rasmussen's paper Khovanov homology and the slice genus, he states as Corollary 3.6 that $s(\mathfrak s_o)=s(\mathfrak s_{\bar o})=s_{min}(K)$, where $s$ is the $s$-grading and $\mathfrak s_o,...

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### Khovanov $A_\infty$ algebra

Let $L$ be a link in $\mathbb{R}^3$, with $D, D'$ be diagrams in
$\mathbb{R}^2$ representing $L$. Khovanov constructed two graded
chain complexes $$C_{D} = (Ch_{D}, d_{D}) \quad C_{D'}=(Ch_{D'},
d_{D'}...

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### A spectral sequence in Khovanov Homology

Szabo equipped the mod $2$ Khovanov complex with a family of differentials $\{d_{i} \}_{i=1}^{\infty}$ such that each $d_{i}$ has bigrading $(i,2i-2)$ where $d_1$ is the mod $2$ Khovanov differential ...

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### Physical Approach to Knot Categorification

Some recent work by Aganagic on knot categorification, Knot Categorification from Mirror Symmetry, Part II: Lagrangians, discusses two categorical approaches to categorification of quantum link ...

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### Khovanov-Rozansky/HOMFLY-PT homology in type B

The picture, as I understand it, is that Hecke algebras (of type A, i.e. associated to the symmetric group $S_n$) lead to the HOMFLY-PT polynomials, and the categorified version of this says that the ...

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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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### What is the "classical limit" of Khovanov homology?

Let me first explain what I mean by the "classical limit".
For quantum group invariants of links and webs (such as colored Jones polynomials), the "classical limit" means the limit $k\rightarrow +\...

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### Version of Khovanov homology that does not produce torsion?

Khovanov homology usualy has torsion. But there are also different versions of Khovanov homology. Is there a Khovanov homology theory that naturaly does not produce torsion?
A followup question: can I ...

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### What is the relationship between the Khovanov-Rozansky homology of a digraph and that of a link?

Motivation: I'm reading this preprint, which takes a digraph $G = (V, E)$ and then builds a projective algebraic set $P(G)$ by assigning a variable to each edge and then defining certain polynomial "...

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### Are there two non-equivalent exotic structures on $\mathbb{R}^4$ coming from topologically slice, non-slice knots?

For a knot $K \subset S^3$, which is topologically slice but not slice (in a smooth way), there's a four manifold $\mathbb{R}^4_K$, homeomorphic but not diffeomorphic to standard euclidean $\mathbb{R}^...

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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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### Questions on poincare homology spheres and branched covers

I have two questions:
Question 1. Suppose that $K$ is a knot in $S^3$. Let $\Sigma(K)$ be the double branched cover of $S^3$ branched along $K$. If $\Sigma(K)=\#_{i=1}^n\Sigma(2,3,5)$, then $K=\#_{i=...

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

Let $K$ be a knot in the 3-sphere $S^3$.
Here we denote by $s(K)$ Rasmussen's s-invariant for $K$,
and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$
by attaching a $2$-...

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### Jones Polynomial as an Equivariant Index

In Khovanov homology the Jones polynomial of a given link is computed as the graded Euler characteristic of a certain complex associated with the link. From other side the Atiyah-Singer theorem ...

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### Understanding a program for computing Khovanov homology

I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp
The ...

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### Khovanov $sl_2$ homology of a connected sum of some torus knots

Let $T_{p,q}$ be the (p,q) torus knot. Could anybody possibly compute either unreduced or reduced Khovanov $\mathfrak{sl}(2)$ homology of the connected sum $T_{2,3} \sharp T_{3,4}$ of the (2,3) and (3,...

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### Is Khovanov's Frobenius algebra self-dual over the integers?

Khovanov's Frobenius algebra (used in the definition of Khovanov homology) is $\mathbb{Z}[X]/X^2$ with the comultiplication. $\Delta(X)=X\otimes X, \Delta(1)=1\otimes X+X\otimes 1$ and the trace $\...

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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, ...

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### Does anyone know this sequence of polynomials?

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 $n\geq 1$}...

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### Is the complete functorial structure for Khovanov--Lee homology known?

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}_{\...