Planar decomposition of the HOMFLY polynomial for bipartite knots and links

TL;DR

This paper introduces a planar decomposition method for HOMFLY polynomials of bipartite knots and links, offering a new approach to their evaluation and potential extensions to homological techniques and other representations.

Contribution

It presents a novel planar calculus for HOMFLY polynomials of bipartite knots, expanding evaluation methods and linking to homological approaches beyond existing arborescent calculus.

Findings

HOMFLY polynomials for bipartite knots have a planar decomposition.

The method applies to various knots, including Kanenobu knots.

It extends to non-fundamental symmetric representations and non-bipartite 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.