Automatic Evaluation of the Links–Gould Invariant for all Prime Knots of up to 10 Crossings

Abstract

This paper describes a method for the automatic evaluation of the Links–Gould two-variable polynomial link invariant (LG) for any link, given only a braid presentation. This method is currently feasible for the evaluation of LG for links for which we have a braid presentation of string index at most 5. Data are presented for the invariant, for all prime knots of up to 10 crossings and various other links. LG distinguishes between these links, and also detects the chirality of those that are chiral. In this sense, it is more sensitive than the well-known two-variable HOMFLY and Kauffman polynomials. When applied to examples which defeat the HOMFLY invariant, interestingly, LG 'almost' fails. The automatic method is in fact applicable to the evaluation of any such state sum invariant for which an appropriate R matrix and cap and cup matrices have been determined.