The 1,2-coloured HOMFLY-PT link homology

Mackaay, Marco; Stošić, Marko; Vaz, Pedro

doi:10.1090/s0002-9947-2010-05155-4

Cited by 31 publications

(44 citation statements)

References 11 publications

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“…So the slightly more general statement of Theorem 14.2 also follows from the proof in [19]. With the invariance under fork sliding (Theorem 13.1) in hand, we can easily prove the invariance of the homotopy type under Reidemeister moves II a , II b , III by an induction using the "sliding bi-gon" method introduced in [31] (and used in [28].) The proof of the invariance under Reidemeister move I is somewhat different and is postponed to the next subsection.…”

Section: Invariance Under Reidemeister Movesmentioning

confidence: 81%

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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: Invariance Under Reidemeister Movesmentioning

confidence: 81%

“…Mackaay, Stosic and Vaz [28] constructed a Z ⊕3 -graded HOMFLYPT homology for 1, 2-colored links, which generalizes Khovanov and Rozansky's construction in [20]. Webster and Williamson [44] further generalized this homology to links colored by any non-negative integers using the equivariant cohomology of general linear groups and related spaces.…”

Section: 3mentioning

confidence: 95%

A colored sl(N) homology for links in S³

2014

Dissertationes Math.

121

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“…We now briefly discuss the case of colored HOMFLY-PT homology. Colored HOMFLY-PT homology was first defined using certain bimodules by Mackaay, Stosic and Vaz [MSV11]. Though they only proved invariance in the case of 1 and 2-colored components, they defined their complexes of bimodules in the case of arbitrary colors.…”

Section: Colored Homfly-pt Homologymentioning

confidence: 99%

Colored Khovanov–Rozansky homology for infinite braids

Abel

Willis

2019

Algebr. Geom. Topol.

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We show that the limiting unicolored sl(N ) Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLY-PT Khovanov-Rozansky homology as well. An application of this result is given in finding a partial isomorphism between the HOMFLY-PT homology of any braid positive link and the stable HOMFLY-PT homology of the infinite torus knot as computed by Hogancamp. arXiv:1709.06666v1 [math.QA] 19 Sep 2017) which is an isomorphism for all homological degrees less than y( ).1.1. Outline of paper. In Section 2 we will review the relevant background needed for colored Khovanov-Rozansky sl(N ) complexes of infinite braids, as well as recalling the results of Cautis from [Cau15]. In Section 3 we restate our main results and corollaries and give a detailed proof. In Section 4 we explore the case of colored Khovanov-Rozansky HOMFLY-PT homology and discuss the estimation theorem for braid positive links.

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“…Furthermore, a categorifaction of colored HOMFLY polynomials P R (K; a, q) as polynomials with two variables (a, q) has led to the triply-graded homology theory called the colored HOMFLY homology H R i,j,k [19][20][21][22][23] whose Poincaré polynomial is called the colored superpolynomial P R (K; a, q, t) = i,j,k…”

Section: Su (N )mentioning

confidence: 99%

Super-A-polynomials for twist knots

Nawata

Ramadevi

Zodinmawia

et al. 2012

J. High Energ. Phys.

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We conjecture formulae of the colored superpolynomials for a class of twist knots K p where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomials for the twist knots with small values of p. The results support the categorified versions of the generalized volume conjecture and the quantum volume conjecture. Furthermore, we obtain the evidence that the Q-deformed Apolynomials can be identified with the augmentation polynomials of knot contact homology in the case of the twist knots.

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The 1,2-coloured HOMFLY-PT link homology

Abstract: Abstract. In this paper we define the 1,2-coloured HOMFLY-PT triply graded link homology and prove that it is a link invariant. We also conjecture on how to generalize our construction for arbitrary colours.

Cited by 31 publications

References 11 publications

A colored sl(N) homology for links in S³

A colored sl(N) homology for links in S³

Colored Khovanov–Rozansky homology for infinite braids

Super-A-polynomials for twist knots

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The 1,2-coloured HOMFLY-PT link homology

Abstract: Abstract. In this paper we define the 1,2-coloured HOMFLY-PT triply graded link homology and prove that it is a link invariant. We also conjecture on how to generalize our construction for arbitrary colours.

Cited by 31 publications

References 11 publications

A colored sl(N) homology for links in S3

A colored sl(N) homology for links in S3

Colored Khovanov–Rozansky homology for infinite braids

Super-A-polynomials for twist knots

Contact Info

Product

Resources

About

A colored sl(N) homology for links in S³

A colored sl(N) homology for links in S³