Some differentials on Khovanov–Rozansky homology

Rasmussen, Jacob

doi:10.2140/gt.2015.19.3031

Cited by 88 publications

(213 citation statements)

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“…In this section, we review the definition and properties of graded matrix factorizations over graded C-algebras, most of which can be found in [19,20,21,36,47]. Some of these properties are stated slightly more precisely here for the convenience of later applications.…”

Section: Graded Matrix Factorizationsmentioning

confidence: 99%

“…Since the Koszul matrix factorizations we use in this paper are more complex than those in [19,20,36,47], it is generally harder to compute them. So it is more important to keep good track of the signs.…”

Section: }mentioning

confidence: 99%

“…Elementary operations on Koszul matrix factorizations. Khovanov and Rozansky [19,20] and Rasmussen [36] introduced several elementary operations on Koszul matrix factorizations that give isomorphic or homotopic graded matrix factorizations. In this subsection, we recall these operations.…”

Section: 4mentioning

confidence: 99%

“…First, note that Lemma 3.12 and Corollary 3.16 are both special cases of Proposition 8.13. Second, recall that Rasmussen [36] explained that the Z 2 -grading of a Koszul matrix factorization can be lifted to a Z-grading. F and G in Proposition 8.13 preserve this Z-grading.…”

Section: Using the Natural Isomorphism Hommentioning

confidence: 99%

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A colored sl(N) homology for links in S³

2014

Dissertationes Math.

121

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Abstract. Fix an integer N ≥ 2. To each diagram of a link colored by 1, . . . , N , we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky in [19]. The homology of this chain complex decategorifies to the Reshetikhin-Turaev sl(N ) polynomial of links colored by exterior powers of the defining representation.

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Section: Graded Matrix Factorizationsmentioning

confidence: 99%

Section: }mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: Using the Natural Isomorphism Hommentioning

confidence: 99%

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A colored sl(N) homology for links in S³

2014

Dissertationes Math.

121

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“…Comparing the constructions is especially interesting in light of the conjecture [3,13] that there should be a spectral sequence from HOMFLY-PT homology to knot Floer homology. In both constructions, a knot in S 3 is studied by considering the collection of graphs G I for I ∈ {0, 1} n obtained by replacing each crossing in an n-crossing braid diagram with its oriented resolution or with a thick edge, as in Figure 1.…”

Section: Introductionmentioning

confidence: 99%

Framed graphs and the non-local ideal in the knot Floer cube of resolutions

Gilmore¹

2015

Algebr. Geom. Topol.

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Abstract. This article addresses the two significant aspects of Ozsváth and Szabó's knot Floer cube of resolutions that differentiate it from Khovanov and Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2) the appearance of a mysterious non-local ideal. Our goal is to facilitate progress on Rasmussen's conjecture that a spectral sequence relates the two knot homologies. We replace the language of twisted coefficients with the more quantum topological language of framings on trivalent graphs. We define a homology theory for framed trivalent graphs with boundary that -for a particular non-blackboard framing -specializes to the homology of singular knots underlying the knot Floer cube of resolutions. For blackboard framed graphs, our theory conjecturally recovers the graph homology underlying the HOMFLY-PT chain complex. We explain the appearance of the non-local ideal by expressing it as an ideal quotient of an ideal that appears in both the HOMFLY-PT and knot Floer cubes of resolutions. This result is a corollary of our main theorem, which is that closing a strand in a braid graph corresponds to taking an ideal quotient of its non-local ideal. The proof is a Gröbner basis argument that connects the combinatorics of the non-local ideal to those of Buchberger's Algorithm.

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Colored Kauffman homology and super-A-polynomials

Nawata

Ramadevi

Zodinmawia

2014

J. High Energ. Phys.

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Abstract:We study the structural properties of colored Kauffman homologies of knots. Quadruple-gradings play an essential role in revealing the differential structure of colored Kauffman homology. Using the differential structure, the Kauffman homologies carrying the symmetric tensor products of the vector representation for the trefoil and the figureeight are determined. In addition, making use of relations from representation theory, we also obtain the HOMFLY homologies colored by rectangular Young tableaux with two rows for these knots. Furthermore, the notion of super-A-polynomials is extended in order to encompass two-parameter deformations of PSL(2, C) character varieties.

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Some differentials on Khovanov–Rozansky homology

Cited by 88 publications

References 20 publications

A colored sl(N) homology for links in S³

A colored sl(N) homology for links in S³

Framed graphs and the non-local ideal in the knot Floer cube of resolutions

Colored Kauffman homology and super-A-polynomials

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Some differentials on Khovanov–Rozansky homology

Cited by 88 publications

References 20 publications

A colored sl(N) homology for links in S3

A colored sl(N) homology for links in S3

Framed graphs and the non-local ideal in the knot Floer cube of resolutions

Colored Kauffman homology and super-A-polynomials

Contact Info

Product

Resources

About

A colored sl(N) homology for links in S³

A colored sl(N) homology for links in S³