Predicting the number and type of twist sites in a rational knot or link

TL;DR

This paper establishes a method to predict the number and type of twist sites in rational knots or links using coefficients from Kauffman's invariant, linking algebraic terms to geometric features.

Contribution

It introduces a way to determine twist site counts and types directly from the coefficients of specific terms in Kauffman's polynomial invariant.

Findings

Coefficient of $z^{c-2}$ equals total twist sites

Coefficients of $a^{-2}z^{c-2}$ and $a^2z^{c-2}$ count left- and right-turning twist sites

Provides a direct algebraic method to analyze knot diagrams

Abstract

A rational knot or link can be put into a standard alternating format which has horizontal and vertical twist sites (double helices). The number and type of these twist sites are determined by terms of next-to-highest $z$-degree in Kauffman's regular isotopy invariant $\Lambda(a,z)$. In particular, for a knot or link with $c$ crossings, the coefficient of the $z^{c-2}$ term is equal to the number of twist sites in its standard diagram. Furthermore, the coefficients of the $a^{-2}z^{c-2}$ and $a^2z^{c-2}$ terms count the number of left-turning and right-turning twist sites, respectively.