A non-torus link from topological vertex

TL;DR

This paper presents the first explicit example of describing a non-torus link, specifically the link L_{8n8}, through the topological vertex formalism, connecting knot invariants with topological string theory.

Contribution

It demonstrates how a non-torus link can be represented via the topological vertex, expanding the applicability of the tangle calculus in topological string theory.

Findings

L_{8n8} link reduces to a Hopf link with composite representations

Resolved conifold with four representations corresponds to a special projection of L_{8n8}

First explicit non-torus link description via topological vertex

Abstract

The recently suggested tangle calculus for knot polynomials is intimately related to topological string considerations and can help to build the HOMFLY-PT invariants from the topological vertices. We discuss this interplay in the simplest example of the Hopf link and link $L_{8n8}$. It turns out that the resolved conifold with four different representations on the four external legs, on the topological string side, is described by a special projection of the four-component link $L_{8n8}$, which reduces to the Hopf link colored with two composite representations. Thus, this provides the first explicit example of non-torus link description through the topological vertex. It is not a real breakthrough, because $L_{8n8}$ is just a cable of the Hopf link, still, it can help to intensify the development of the formalism towards more interesting examples.