Positive knots and ribbon concordance

JOE BONINGERRibbon 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 are concordant and | (K )| 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. The Alexander polynomial K 0 of K 0 divides K 1 [12; 14]. The genus g(K 0 ) of K 0 is less than or equal to g(K 1 ) [42]. If K 1 is fibered, then K 0 is as well [25; 42].