Positive Knots and Ribbon Concordance

TL;DR

This paper proves that positive knots are minimal in the ribbon concordance partial order for a large class, and shows that positive knots with certain properties are concordant only if their Alexander modules are isomorphic, supporting uniqueness conjectures.

Contribution

It extends previous results by proving minimality of positive knots in a broad class and relates concordance to Alexander modules, advancing understanding of positive knot classification.

Findings

Positive knots are minimal in the ribbon concordance order for a large class.

Positive knots with certain signature and genus conditions have isomorphic rational Alexander modules if concordant.

Positive knots cannot be expressed as non-trivial band sums.

Abstract

Ribbon concordances between knots generalize the notion of ribbon knots. Agol, building on work of Gordon, proved ribbon concordance gives a partial order on knots in $S^3$. In previous work, the author and Greene conjectured that positive knots are minimal in this ordering. In this note we prove this conjecture for a large class of positive knots, and show that a positive knot cannot be expressed as a non-trivial band sum -- both results extend earlier theorems of Greene and the author for special alternating knots. In a related direction, we prove that if positive knots $K$ and $K'$ are concordant and $|\sigma(K)| \geq 2g(K) - 2$, then $K$ and $K'$ 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.