Are there p-adic knot invariants?

TL;DR

This paper proposes a p-adic extension of knot invariants using Hall-Littlewood polynomials, aiming to define p-adic HOMFLY-PT polynomials for torus knots, which could influence various areas of modern mathematical theory.

Contribution

It introduces a novel approach to defining p-adic knot invariants via Hall-Littlewood polynomials, extending the Rosso-Jones formula to p-adic contexts for torus knots.

Findings

Proposes p-adic HOMFLY-PT polynomials for torus knots

Establishes topological invariance under certain symmetries

Suggests potential for generalizations to other knot families

Abstract

We suggest to use the Hall-Littlewood version of Rosso-Jones formula to define the germs of $p$-adic HOMFLY-PT polynomials for torus knots $[m,n]$, which possess at least the $[m,n] \longleftrightarrow [n,m]$ topological invariance. This calls for generalizations to other knot families and is a challenge for several branches of modern theory.