The Links-Gould invariants as generalizations of the Alexander polynomial

Abstract : In this thesis we focus on the connections that exist between two link invariants: first the Alexander-Conway invariant ∆ that was the first polynomial link invariant to be discovered, and one of the most thoroughly studied since alongside with the Jones polynomial, and on the other hand the family of Links-Gould invariants LGn,m that are quantum link invariants derived from super Hopf algebras Uqgl(n|m). We prove a case of the De Wit-Ishii-Links conjecture: in some cases we can recover powers of the Alexander polynomial as evaluations of the Links-Gould invariants. So the LG polynomials are generalizations of the Alexander invariant. Moreover we give evidence that these invariants should still have some of the most remarkable properties of the Alexander polynomial: they seem to offer a lower bound for the genus of links and a criterion for fiberedness of knots.
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Ben-Michael Kohli. The Links-Gould invariants as generalizations of the Alexander polynomial. General Topology [math.GN]. Université de Bourgogne, 2016. English. ⟨NNT : 2016DIJOS062⟩. ⟨tel-01722913⟩

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