Evolution method and HOMFLY polynomials for virtual knots

Abstract

Following the suggestion of arXiv:1407.6319 to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel 2-strand links. Within this family one can check topological invariance and understand how differential hierarchy is modified in virtual case. This opens a way towards a definition of colored (not only cabled) knot polynomials, though problems still persist beyond the first symmetric representation.

28 pages

References in corpus (21)

• Khovanov's homology for tangles and cobordisms
• On Khovanov's categorification of the Jones polynomial
• Matrix factorizations and link homology II
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• sl(3) link homology
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links III : Compact Semi-simple Group
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix
• On the Hecke algebras and the colored HOMFLY polynomial
• Minimal surface representations of virtual knots and links
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux
• Differential hierarchy and additional grading of knot polynomials
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Colored Kauffman Homology and Super-A-polynomials
• Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
• Link polynomial calculus and the AENV conjecture
• Invariants of Links in Thickened Surfaces
• On possible existence of HOMFLY polynomials for virtual knots
• The parity hierarchy and new invariants of knots in thickened surfaces

Cited by in corpus (8)

• Tabulating knot polynomials for arborescent knots
• Colored knot polynomials. HOMFLY in representation [2,1]
• Are there p-adic knot invariants?
• On matrix-model approach to simplified Khovanov-Rozansky calculus
• On the block structure of the quantum R-matrix in the three-strand braids
• Towards formalization of the soliton counting technique for the Khovanov-Rozansky invariants in the deformed $\mathcal{R}$-matrix approach
• Matrix model and dimensions at hypercube vertices
• On ambiguity in knot polynomials for virtual knots