Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials

TL;DR

This paper introduces new obstructions to homotopy ribbon concordance using Blanchfield forms and Levine-Tristram signatures, and demonstrates the existence of infinite knot families sharing certain invariants but not being homotopy ribbon concordant.

Contribution

It develops novel homotopy ribbon concordance obstructions from algebraic invariants and constructs infinite knot families with identical invariants yet distinct concordance properties.

Findings

Obstructions from Blanchfield form and Levine-Tristram signatures are effective.

Existence of infinite knot families with same Alexander polynomial and Blanchfield form.

No pair in these families is homotopy ribbon concordant.

Abstract

We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine-Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there exists an infinite family of knots that are all concordant to K and have the same Blanchfield form as K, such that no pair of knots in that family is homotopy ribbon concordant.