Invariants of rational links represented by reduced alternating diagrams

TL;DR

This paper develops an algorithm to convert nonalternating continued fractions of rational links into all-even form, enabling new formulas for braid index and HOMFLY polynomial based on reduced alternating diagrams.

Contribution

It introduces an algorithmic transformation from nonalternating to all-even continued fractions, facilitating formulas for key invariants of rational links.

Findings

Derived formulas for braid index using reduced alternating diagrams.

Established methods to compute HOMFLY polynomial from nonalternating forms.

Provided an algorithmic approach for transforming continued fractions.

Abstract

A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a {\em nonalternating form} and the diagram corresponding to it is a reduced alternating link diagram, which is minimum in terms of the number of crossings in the diagram. Famous formulas exist in the literature for the braid index of a rational link by Murasugi and for its HOMFLY polynomial by Lickorish and Millet, but these rely on a special continued fraction expansion of the rational number in which all partial denominators are even (called {\em all-even form}). In this paper we present an algorithmic way to transform a continued fraction given in nonalternating form into the all-even form. Using this method we derive formulas for the braid index and the HOMFLY polynomial of a rational link in terms of its reduced alternating form, or equivalently the nonalternating form of the corresponding rational number.