A note on the Lickorish-Millett-Turaev formula for the Kauffman polynomial

TL;DR

This paper provides a concise proof of the Lickorish-Millett-Turaev formula for the Kauffman polynomial at a specific parameter value, using a novel approach involving expressing nonoriented links as sums of oriented links.

Contribution

It introduces a new proof technique that clarifies the formula's connection to linking numbers, enhancing understanding of the Kauffman polynomial's properties.

Findings

The proof simplifies understanding of the Lickorish-Millett-Turaev formula.

It reveals the formula as a generating function for linking numbers.

The approach explains the formula's combinatorial structure.

Abstract

We use the idea of expressing a nonoriented link as a sum of all oriented links corresponding to the link to present a short proof of the Lickorish-Millett-Turaev formula for the Kauffman polynomial at $z= -a- a^{-1}$. Our approach explains the observation made by Lickorish and Millett that the formula is the generating function for the linking number of a sublink of the given link with its complementary sublink.