Homotopy ribbon concordance and Alexander polynomials

Abstract

We show that if a link J in the 3-sphere is homotopy ribbon concordant to a link L, then the Alexander polynomial of L divides the Alexander polynomial of J.

Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials

“…We provide an obstruction for the homotopy ribbon concordance in terms of twisted Alexander modules, which generalizes the results in [FP19] on the classical Alexander module.…”

“…The Alexander polynomial ∆ J = (t 2 − t + 1) 2 (t 2 − 3t + 1) is divisible by the Alexander polynomial ∆ K = (t 2 − t + 1) 2 , and so ∆ J = ∆ 2 J is divisible by ∆ K = ∆ 2 K . Thus the known Alexander polynomial homotopy ribbon obstruction [FP19] is not applicable for J and K. Moreover, one may check by computer calculation that there exist bigraded injections H(K) → H(J) for H either Khovanov homology, or knot Floer homology. So the ribbon concordance obstructions of Khovanov homology [LZ19] and knot Floer homology [Z19] are not applicable for J and K, either.…”

“…We study homotopy ribbon concordance of knots, extending the work [FP20] by the first and fifth authors on the classical Alexander polynomial to the Blanchfield form and to Levine–Tristram signatures. Then, we present an application of twisted Alexander polynomials to homotopy ribbon concordance.…”

“…Then, we present an application of twisted Alexander polynomials to homotopy ribbon concordance. As described in [FP20], the classical Alexander polynomial is useful to show that there exists an infinite family of concordant knots that are not homotopy ribbon concordant to each other. In this paper, we show that there exists such an infinite family of knots even with isomorphic Seifert forms, and so also isomorphic Blanchfield forms, using twisted Alexander polynomials.…”

Critical point counts in knot cobordisms: abelian and metacyclic invariants

“…The following corollary is immediate. For related results in the case of ribbon concordances, see [7].…”