On the Links-Gould invariant and the square of the Alexander polynomial (Q2796946)

The Links-Gould invariants \(LG^{m,n}(L; t_0, t_1)\) and the Alexander-Conway invariant \(\Delta_L(t_0)\) are both invariants of oriented links \(L\in S^3\). In [\textit{D. De Wit} et al., Algebr. Geom. Topol. 5, 405--418 (2005; Zbl 1079.57004)], it was conjectured that \(LG^{m,n}(L; t_0, e^{2i\pi/n}t_0^{-1})=\Delta_L(t_0^n)^m\) and the conjecture was proved in the case \(m=1\) there.NEWLINENEWLINEThis paper proves the conjecture for \(m=2, 3\) and \(n=1\). Oriented links can be presented as closures of braids so some braid group representations give rise to oriented link invariants. The Alexander-Conway invariant can be derived from the Burau representation \(F\) of the braid group. While the Links-Gould invariants \(LG^{2,1}\) and \(LG^{3,1}\) can be derived from braid group representations that are, as proved in this paper, isomorphic to \(\wedge(F\oplus F)\) and \(\wedge(F\oplus F\oplus F)\), respectively. The relations in the braid group representations lead to the proof of the conjecture in these two cases.