Computing Khovanov–Rozansky homology and defect fusion

Abstract

We compute the categorified sl(N ) link invariants as defined by Khovanov and Rozansky, for various links and values of N . This is made tractable by an algorithm for reducing tensor products of matrix factorisations to finite rank, which we implement in the computer algebra package Singular.2000 Mathematics Subject Classification 57M27 1 We do not use the (representation theoretically more appropriate) "wide edge" depiction of [KR08a] for the second diagram. This is done to facilitate the interpretation in terms of Landau-Ginzburg models below. Nils Carqueville

HOMFLY polynomials in representation [3, 1] for 3-strand braids

“…This is cured in a far less trivial "differential expansion" [63,83,84,98], which also reflects the hidden "differential structure" [48] lying in the base of the Khovanov approach [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] to knot polynomials, very different from the R-matrix one exploited in the present paper. In variance with naive genus expansion, the differential expansion contains only a finite number of terms up to the r + s power of Z's, where Z (k)…”

Towards formalization of the soliton counting technique for the Khovanov–Rozansky invariants in the deformed ℛ-matrix approach

“…Each of the turning-points is treated as a two-valent vertex on the knot diagram, of type ■ , ✠ , ✒ or | . Table 3: Summary of the discussed approaches, with bibliographic and internal references Khovanov polynomial N = 2 Introduced in [5] Computational technique developed in [18] Table of results, together with computer code [7] Presented for all prime knots (up to 11 crossings) and links (up to 11 crossings); in principle, computed for any knots Reviewed, e.g., in [19,20], [21,9] Khovanov-Rozansky polynomial N ∈ Z + The definition introduced in [4] Applied to explicit computations in [11] "Thin" knots up to 9 crossings [22] Knots and links up to 6 crossings, mostly for particular vales of N The appoach reviewed, e.g., in [21,9] Attemts of modification Tensor-like formalism [23] Simplest examples, 2-strand torus knots, twist knots R-matrix bases formalism [3] 2 and 3-strand torus knots, 3-and 4-strand knots and links up to 6 crossings, twocomponent links from two antiparallel strands ssec. 2,4 Positive division technique [3] ssec.…”

A Recursive Formula for the Khovanov Cohomology of Kanenobu Knots

“…The khovanov cohomology groups of the Kanenobu knot K(0, 0) are given by the following. (4,9), (−4, −7), (−4, −9)}; otherwise.…”

“…Then how to calculate these cohomology? In [4], N. Carqueville and D. Murfet presented the first method to directly compute HKR N for arbitrary links. Therefore, it might be feasible to give a formula to HKR N (K(p, q)).…”