The Links-Gould invariant of oriented links is a two-variable polynomial invariant constructed from a one-parameter family of the four dimensional representations of the quantum superalgebra \(U_q[gl(m| n)]\) [J. R. Links and M. D. Gould, Lett. Math. Phys. 26, No. 3, 187–198 (1992; Zbl 0777.57005)]. This invariant can detect the chirality of some links where the HOMFLY and Kauffman polynomials fail, namely the Links-Gould invariant is distinct from the HOMFLY and Kauffman polynomials. On the other hand, the Links-Gould invariant does not distinguish mutants or inverses [D. de Wit, J. R. Links and L. H. Kauffman, J. Knot Theory Ramifications 8, No. 2, 165–199 (1999; Zbl 0934.57005)].
In the paper under review, the authors reveal the existence of infinitely many prime knots and links which have no mutants and cannot be distinguished by the Links-Gould invariant. Actually they show that there exist arbitrarily many \(2\)-bridge knots and links which share the same Links-Gould invariant by taking advantage of the second author’s example of arbitrarily many \(2\)-bridge knots (resp. links) which share the same HOMFLY and Kauffman (resp. \(2\)-variable Alexander) polynomials [Yokohama Math. J. 38, No. 2, 145–154 (1991; Zbl 0744.57006)]. As an application, they give an example of a chiral hyperbolic knot whose chirality cannot be detected by the Links-Gould invariant. A relationship between the Links-Gould invariant and the Conway polynomial for links is also given. As an application of the relation, they show that there exist arbitrarily many \(2\)-bridge knots which share the same Kauffman polynomial but can be distinguished completely by the Links-Gould invariant.