On the skein polynomial for links

TL;DR

This paper provides new characterizations of the skein polynomial for links, including Jones and Alexander-Conway polynomials, without relying on the traditional crossing smoothing move, and extends these characterizations to knots and links with any number of components.

Contribution

It introduces alternative characterizations of the skein polynomial and related invariants that do not depend on the smoothing move, broadening understanding of these link polynomials.

Findings

Characterizations of the skein polynomial avoiding crossing smoothing

Characterizations of Jones and Alexander-Conway polynomials derived from the skein polynomial

Extensions of characterizations to knots and links with arbitrary components

Abstract

We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual "smoothing of a crossing" move. As by-products we have characterizations of these polynomials for knots, and for links with any given number of components.