The Links–Gould Invariant as a Classical Generalization of the Alexander Polynomial?

Abstract

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.

On the Links–Gould invariant and the square of the Alexander polynomial

“…In this paper we prove inequality (1.2) that had been conjectured in [9]. In particular, (1.1) shows that the bound obtained that way systematically improves the classical lower bound for the genus of a knot given by the Alexander invariant.…”

“…As it is explained in [9], Proposition 1.13, a consequence of this inequality and of [14] is the following.…”

“…(1.2) deg(LG(K, t 0 , t 1 )) ≤ 4g(K). This is one of many features of the Links-Gould invariant, proven or conjectured, that illustrate the fact that this quantum invariant has deep topological meaning (see in particular [7,9]). In general, it is hard to deduce precise topological properties on a knot or link from the value quantum invariants take on that link.…”

Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

“…[14]). The structure on the right-hand side of (3.5) also appears in the literature on generalized Alexander invariants and can be viewed as a TQFT reason why such generalizations often end up related to the Alexander polynomial [62][63][64][65]. (See, however, the discussion in Section 4).…”

Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

“…[14]). The structure on the right-hand side of (3.5) also appears in the literature on generalized Alexander invariants and can be viewed as a TQFT reason why such generalizations often end up related to the Alexander polynomial [60][61][62][63]. (See, however, the discussion in Section 4).…”