The cobordism distance between a knot and its reverse

Abstract

We consider the question of how knots and their reverses are related in the concordance group C. There are examples of knots for which K = K r ∈ C. This paper studies the cobordism distance, d(K, K r ). If K = K r ∈ C, then d(K, K r ) > 0 and it elementary to see that for all K, d(K, K r ) ≤ 2g 4 (K). It is known that d(K, K r ) can be arbitrarily large. Here we present a proof that for non-slice knots satisfying g 3 (K) = g 4 (K), one has d(K, K r ) ≤ 2g 4 (K) − 1. This family includes all strongly quasi-positive knots and all non-slice genus one knots. We also construct knots K of arbitrary four-genus for which d(K, K r ) = g 4 (K). Finding knots for which d(K, K r ) > g 4 (K) remains an open problem.