Where the Links--Gould invariant first fails to distinguish nonmutant prime knots

Abstract

It is known that the first two-variable Links--Gould quantum link invariant $LG\equiv LG^{2,1}$ is more powerful than the HOMFLYPT and Kauffman polynomials, in that it distinguishes all prime knots (including reflections) of up to 10 crossings. Here we report investigations which greatly expand the set of evaluations of $LG$ for prime knots. Through them, we show that the invariant is complete, modulo mutation, for all prime knots (including reflections) of up to 11 crossings, but fails to distinguish some nonmutant pairs of 12-crossing prime knots. As a byproduct, we classify the mutants within the prime knots of 11 and 12 crossings. In parallel, we learn that $LG$ distinguishes the chirality of all chiral prime knots of at most 12 crossings. We then demonstrate that every mutation-insensitive link invariant fails to distinguish the chirality of a number of 14-crossing prime knots. This provides 14-crossing examples of chiral prime knots whose chirality is undistinguished by $LG$.