A colored sl(N) homology for links in S³

Abstract

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.

3d-3d correspondence revisited

“…Namely, the flat connection on the solid surgery torus with eigenvalues (−1, −1) is trivial, while the flat connection on the trefoil knot complement with eigenvalues (−1, −1) is parabolic, meaning the full holonomy matrix is −1 1 0 −1 . This is not unexpected, since (x, y) = (−1, −1) lies on the nonabelian branch y + x 6 = 0 of 12 In contrast to the rest of the paper, we take care in this section to write A-polynomials in terms of actual SL(2, C) meridian and longitude eigenvalues rather than their squares. Thus, for the trefoil, the nonabelian A-polynomial is written as y + x 6 rather than y + x 3 .…”

“…Therefore, at least formally, Note that the three terms (s) − ∞ , (−1/(qst)) − ∞ , and (x/s) − ∞ each contribute a half-line of poles to Υ 3 1 . If we take q > 1 to be real, then the asymptotics of the integrand are given by 12) so the integral along Γ I does converge in a suitable range of x and t (namely, if |xt 3 | < 1). In contrast, the integrals along the other obvious cycles here, Γ II and Γ III , always converge.…”

“….) stands for the Poincaré polynomial of a doubly-graded [9][10][11][12] or triply-graded [13][14][15] homology theory…”

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

“…A similar approach is taken in [24] with the coloured versions. Ideally we would like to be able to do the same thing here, but MOY moves for the Alexander polynomial have minus signs that do not appear in the sl.n/ case, which suggests any categorification takes a different flavour to the categorifications of sl.n/ polynomials, as the minus signs suggest the use of chain complexes in a way that was not necessary in those cases.…”

Lectures on Knot Homology and Quantum Curves

“…Much like the Khovanov homology of a knot is a categorification of its Jones polynomial or quantum sl(2) invariant, there exist generalizations [57,56,48,8,20] of the Khovanov homology categorifying the n-colored Jones polynomials for all n:…”