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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 Kauffmanpolynomials, in that it distinguishes all prime knots (including reflections)of up to 10 crossings. Here we report investigations which greatly expand theset of evaluations of $LG$ for prime knots. Through them, we show that theinvariant is complete, modulo mutation, for all prime knots (includingreflections) of up to 11 crossings, but fails to distinguish some nonmutantpairs of 12-crossing prime knots. As a byproduct, we classify the mutantswithin 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 todistinguish the chirality of a number of 14-crossing prime knots. This provides14-crossing examples of chiral prime knots whose chirality is undistinguishedby $LG$.

WHERE THE LINKS–GOULD INVARIANT FIRST FAILS TO DISTINGUISH NONMUTANT PRIME KNOTS