Planar decomposition of the HOMFLY polynomial for bipartite knots and links

Abstract

The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but for a special class of bipartite diagrams made entirely from the anitparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters -- and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov-Rozansky generalisations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links.

arXiv comment33 pages, published version

References in corpus (27)

• 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
• Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix
• Evolution method and "differential hierarchy" of colored knot polynomials
• Asymptotics of the colored Jones function of a knot
• New Quantum Obstructions to Sliceness
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Comments on the Casimir energy in supersymmetric field theories
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
• Factorization of differential expansion for antiparallel double-braid knots
• From Topological to Quantum Entanglement
• On rectangular HOMFLY for twist knots
• Introduction to Khovanov Homologies. II. Reduced Jones superpolynomials
• Differential expansion for link polynomials
• Perspectives of differential expansion
• Extension of KNTZ trick to non-rectangular representations
• Knot polynomials for twist satellites
• On possible existence of HOMFLY polynomials for virtual knots
• Factorization of differential expansion for non-rectangular representations
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams
• Towards tangle calculus for Khovanov polynomials
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• On measuring the topological charge of anyons

Cited by in corpus (8)

• Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
• Bipartite expansion beyond biparticity
• On geometric bases for quantum A-polynomials of knots
• Khovanov-Rozansky cycle calculus for bipartite links
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs
• Khovanov--Rozansky matrix factorization reduction for bipartite links
• Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials