DAHA approach to iterated torus links

TL;DR

This paper extends DAHA-Jones polynomials and superpolynomials from knots to links, including algebraic links, using Macdonald polynomials and splice diagrams, and explores their connections to HOMFLY-PT and Khovanov-Rozansky polynomials.

Contribution

It introduces a DAHA-based construction for iterated torus links, generalizing previous knot-focused work to links and establishing new links to topological and algebraic invariants.

Findings

Construction produces DAHA-vertex for Hopf links

Specialization t=q conjecturally yields HOMFLY-PT polynomials

Provides partial confirmations of relations to Khovanov-Rozansky and ORS-polynomials

Abstract

We extend the construction of the DAHA-Jones polynomials for any reduced root systems and DAHA-superpolynomials in type A from the iterated torus knots (our previous paper) to links, including arbitrary algebraic links. Such a passage essentially corresponds to the usage of the products of Macdonald polynomials and is directly connected to the so-called splice diagrams. The specialization t=q of our superpolynomials conjecturally results in the HOMFLY-PT polynomials. The relation of our construction to the stable Khovanov-Rozansky polynomials and the so-called ORS-polynomials of the corresponding plane curve singularities is expected for algebraic links in the uncolored case. These 2 connections are less certain, since the Khovanov-Rozansky theory for links is not sufficiently developed and the ORS polynomials are quite involved. However we provide some confirmations. For Hopf links, our construction produces the DAHA-vertex, similar to the refined topological vertex, which is an important part of our paper.