In this paper we conjecture that the Links-Gould invariant of links, that we know is a generalization of the Alexander-Conway polynomial, shares some of its classical features. In particular it seems to give a lower bound for the genus of links and to provide a criterion for beredness of knots. We give some evidence for these two assumptions. Contents Introduction 1 1. The Links-Gould invariant and the genus of links 2 2. Evidence supporting the genus conjecture 5 3. The Links-Gould polynomial and beredness 15 4. Appendix : proof of the harder case in Theorem 2.23 17 References 20 2010 Mathematics Subject Classication. 57M27.

This paper gives a connection between well-chosen reductions of the Links–Gould invariants of oriented links and powers of the Alexander–Conway polynomial. This connection is obtained by showing the representations of the braid groups we derive the specialized Links–Gould polynomials from can be seen as exterior powers of a direct sum of Burau representations.

Oleg Viro studied in [14] two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to Uqsl(2) at fourth roots of unity, or by considering the super Hopf algebra Uqgl(1|1). In this paper, we show these Hopf algebras share properties with the −1 specialization of Uqgl(n|1) leading to the proof of a conjecture of David De Wit, Atsushi Ishii and Jon Links on the Links-Gould invariants.