Some differentials on Khovanov–Rozansky homology

By Jacob Rasmussen. 2006

172 citations recorded by the source corpus.

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 KR–homologies and the relations between them. Our motivation was to give some substance to the conjectures made in Dunfield, Gukov and Rasmussen [3] about the behavior of these theories and their relation to the knot Floer homology. Although we are unable to say anything about the latter problem, we hope that we can at least shed some light on the structure of KR–homology.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/15c59e33e6032e45c88ddda9d7b2b312a484501d

How researchers see this paper

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In Discrete Mathematics and Combinatorics

1. Andrei Neguct
2. E. Gorsky
3. P. Kronheimer
4. T. Mrowka
5. I. Danilenko
6. Jacob Rasmussen
7. David E. V. Rose
8. Paul Wedrich

In Geometry and Topology

1. T. Mrowka
2. P. Kronheimer
3. E. Gorsky
4. Jacob Rasmussen
5. Andrei Neguct
6. I. Danilenko
7. I. Cherednik
8. David E. V. Rose

In Mathematical Physics

1. T. Mrowka
2. P. Kronheimer
3. Jacob Rasmussen
4. I. Cherednik
5. E. Gorsky
6. Andrei Neguct
7. Vivek V. Shende
8. Paul Wedrich

Citations in the corpus

• Super $q$-Howe duality and web categories (2015, unknown): “Using Lemma 2.25 and additionally [24, Section 4.2], where Rasmussen’s singular crossings correspond to green dumbbells with label 2 , provides the statement.”
• DAHA and iterated torus knots (2014, background): “This is generally provided by the theory of differentials @nC1 from [26; 37].”
• The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, methodology): “This has several slight variants; we discuss what is called by Rasmussen [60] the unreduced homology, and denoted by H i;j;k .”
• Deformations of colored sl(N) link homologies via foams (2015, background): “Lobb [Lo1, Lo2] has used Gornik’s generic deformation of slN link homology to define concordance invariants that are analogous to Rasmussen’s invariant.”
• Jones polynomials of torus knots via DAHA (2011, background): “Using [24, 25], Rasmussen [33] established the existence of such differentials and conjectured the degeneracy of the corresponding spectral sequence, which guarantees the desired relation to the Khovanov–Rozansky polynomials with the reservation that the differentials can be very difficult to calcul”
• Knot homology groups from instantons (2008, background): “The situation is reminiscent of the large-N stabilization for Khovanov-Rozansky homology that is discussed in [10, 37].”
• AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “They were constructed by Rasmussen in [Ras15], and correspond to the d 1 and d − 1 differentials conjectured in [DGR06].”
• Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology (2016, methodology): “Recall that in [49] the third author constructed a spectral sequence from the HOMFLY-PT homology to the glN homology of any knot.”
• On the computation of torus link homology (2016, methodology): “ by Khovanov [Kho07a] using the Hochschild homology of Soergel bimodules. It has generated a great deal of interest, admitting spectral sequences which converge to various sl n-link homology theories [Ras15], and having deep connections to the representation theory of Hecke algebras in type A. Khova”