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 ≡ 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 thatLG 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. The Links-Gould invariantFor any positive integers m and n, the Links-Gould quantum link invariant LG m,n is a two-variable invariant of oriented links. We here describe it using the variables q and p ≡ q α+(m−n)/2 which are inherited from its definition via the 2 mn -dimensional α-parametric representation of highest weight (0 m |α n ) of the quantum superalgebra U q [gl(m|n)]. Its construction was originally described in [12,25], and some of its properties together with some concrete evaluations are available in [5,6,8,9].The case LG 1,1 , in which the two variables degenerate to a single variable (p), is actually the Alexander-Conway polynomial, and the next simplest case, LG 2,1 , is the first truly two-variable invariant of the family. Apart from LG 1,1 , more is known about LG 2,1 than about any of its kin [4,[15][16][17][18][19][20][21], and we generally refer to LG 2,1 as the Links-Gould invariant LG. In particular, it has been proven to be polynomial [17]; this is still only surmised for general LG m,n .

The Links–Gould Invariant as a Classical Generalization of the Alexander Polynomial?

“…The reason why we did not test all non alternating 13 crossing prime knots is explained in [6] : The LINKS-GOULD EXPLORER's database contains evaluations only for LG of knots with string index at most 5, and from time to time 6 or 7. Indeed, the memory required increases dramatically with braid width.…”

The Links–Gould polynomial as a generalization of the Alexander–Conway polynomial

“…We remark that, because of the symmetry (t) • = (t −1 ), the Alexander-Conway polynomial does not detect the chirality of knots. In contrast, the LG polynomial does detect the chirality of knots, at least for all prime knots with up to 12 crossings [De Wit and Links 2005].…”

Tabulating and Distinguishing Mutants

“…At an intermediate stage, when I had done this for 11 and 12 crossing knots, D. De Wit informed me that he had a similar tabulation done mostly "by hand" in [8]. (He had, at that time, not published it completely, so I verified only that the number of groups coincided.…”