# sl3–foam homology calculations

@article{Lewark2012sl3foamHC, title={sl3–foam homology calculations}, author={Lukas Lewark}, journal={Algebraic \& Geometric Topology}, year={2012}, volume={13}, pages={3661-3686} }

We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots, which are counterexamples to Lobb’s conjecture that the sl3 ‐knot concordance invariant s3 (suitably normalised) should be equal to the Rasmussen invariant s2 . For this family,js3j js2j. The main tool is an implementation of Morrison and Nieh’s algorithm to calculate Khovanov’s sl3 ‐foam link homology. Our C++ program is fast enough to calculate the integral homology of, eg, the .6;5/‐torus knot in six…

## 12 Citations

### On Stable -Homology of Torus Knots

- MathematicsExp. Math.
- 2015

The stable Khovanov– Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit irregular sequence of polynomials. We verify this conjecture using newly…

### Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?

- Mathematics
- 2018

A bstractR-coloured knot polynomials for m-strand torus knots Torus[m,n] are described by the Rosso-Jones formula, which is an example of evolution in n with Lyapunov exponents, labelled by Young…

### New quantum obstructions to sliceness

- Mathematics
- 2015

It is well known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussen's concordance invariant s . This gives a concordance…

### A CLOSED FORMULA FOR THE EVALUATION OF slN -FOAMS

- Mathematics
- 2018

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the slN link…

### Computer Bounds for Kronheimer–Mrowka Foam Evaluation

- MathematicsExperimental Mathematics
- 2021

Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. Their approach is based on a functor…

### A closed formula for the evaluation of $\mathfrak{sl}_N$-foams

- Mathematics
- 2017

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the…

### Categorification of the colored $\mathfrak{sl}_3$-invariant

- Mathematics
- 2015

We give explicit resolutions of all finite dimensional, simple $U_q(\mathfrak{sl_3})$-modules. We use these resolutions to categorify the colored $\mathfrak{sl}_3$-invariant of framed links via a…

### 𝔤𝔩n-webs, categorification and Khovanov–Rozansky homologies

- MathematicsJournal of Knot Theory and Its Ramifications
- 2020

In this paper, we define an explicit basis for the [Formula: see text]-web algebra [Formula: see text] (the [Formula: see text] generalization of Khovanov’s arc algebra) using categorified [Formula:…

### CATEGORIFICATION OF THE COLORED sl 3 -INVARIANT

- Mathematics
- 2015

. We give explicit resolutions of all ﬁnite dimensional, simple U q ( sl 3 )-modules. We use these resolutions to categorify the colored sl 3 -invariant of framed links via a complex of complexes of…

## References

SHOWING 1-10 OF 48 REFERENCES

### The universal sl3–link homology

- Mathematics
- 2007

We define the universal sl3 –link homology, which depends on 3 parameters, following Khovanov’s approach with foams. We show that this 3–parameter link homology, when taken with complex coefficients,…

### Computations of the OzsvthSzab knot concordance invariant

- Mathematics
- 2004

Ozsvath and Szabo have defined a knot concordance invariantthat bounds the 4-ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots…

### Heegaard Floer homology and alternating knots.

- Mathematics
- 2003

In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y , which is closely related to the Heegaard Floer homology of Y . In this paper we…

### On Khovanov's cobordism theory for SU3 knot homology

- Mathematics
- 2006

We reconsider the link homology theory defined by Knovanov in [9] and generalized by Mackaay and Vaz in [15]. With some slight modifications, we describe the theory as a map from the planar algebra…

### Knot Floer homology and the four-ball genus

- Mathematics
- 2003

We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance…

### Triply-graded link homology and Hochschild homology of Soergel bimodules

- Mathematics
- 2005

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of…

### On the Khovanov and knot Floer homologies of quasi-alternating links

- Mathematics
- 2007

Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer…

### Some differentials on Khovanov–Rozansky homology

- Mathematics
- 2006

In [12; 13], Khovanov and Rozansky introduced a new class of homological knot invariants which generalize the original construction of the Khovanov homology [9]. In this paper, we investigate these…

### A note on Gornik's perturbation of Khovanov-Rozansky homology.

- Mathematics
- 2012

We show that the information contained in the associated graded vector space to Gornik’s version of Khovanov‐Rozansky knot homology is equivalent to a single even integer sn.K/. Furthermore we show…

### Torsion of the Khovanov homology

- Mathematics
- 2004

Khovanov homology is a recently introduced invariant of oriented links in $\mathbb{R}^3$. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov…