Positive Knots and Ribbon Concordance

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.