Positive knots and ribbon concordance

Joe Boninger

Vol. 335 (2025), No. 1, 81–95

DOI: 10.2140/pjm.2025.335.81

Abstract

Ribbon concordances between knots generalize the notion of ribbon knots. Agol (2022) proved ribbon concordance which gives a partial order on knots in $S^3$, and Boninger and Greene (2024) conjectured that positive knots are minimal in this ordering. In this article we prove this conjecture for a large class of positive knots, and show that a positive knot cannot be expressed as a nontrivial band sum. Both results extend earlier theorems of Boninger and Greene for special alternating knots. In a related direction, we prove that if positive knots $K$ and $K^{\prime }$ are concordant and $|\sigma (K)|\ge 2g(K)-2$, then $K$ and  $K^{\prime }$ have isomorphic rational Alexander modules. This strengthens a result of Stoimenow, and gives evidence toward a conjecture that any concordance class contains at most one positive knot.

Keywords

geometric topology, knot theory, concordance, ribbon concordance, positive knots

Mathematical Subject Classification

Primary: 57K10

Milestones

Received: 10 November 2024

Revised: 6 February 2025

Accepted: 4 March 2025

Published: 8 April 2025