The Links–Gould Invariants of links as classical generalizations of the Alexander polynomial

Abstract 

The Links-Gould invariants of links are a family of two variable polynomial quantum link invariants built using Hopf superalgebras U_q(gl(m|n)).

However, we now know that the Alexander invariant of links can be recovered by evaluating the Links-Gould invariants in several different ways, using works by De Wit-Ishii-Links, and more recently Kohli and Patureau-Mirand. Therefore, all the information that the Alexander polynomial carries is also contained in the Links-Gould invariants of links. That includes the homological information that the Alexander invariant gives about the link: genus bound, fiberdness, …

Hence, one can wonder whether the Links-Gould invariants of links could generalize some of the classical properties of the

Alexander polynomial. It seems that it should be the case, and recent work with Geer, Patureau-Mirand and Tahar hints further to a classical construction for these quantum invariants.