The cobordism distance between a knot and its reverse

TL;DR

This paper investigates the cobordism distance between a knot and its reverse, revealing bounds and specific cases where this distance is less than twice the four-genus, with implications for knot theory.

Contribution

It demonstrates that for knots with equal four- and three-genus, the cobordism distance to their reverse is strictly less than twice the four-genus, and constructs examples where this distance equals the genus.

Findings

d(K,K^r) < 2g_4(K) for certain knots

Existence of knots with d(K,K^r) = g = g_4(K)

No known examples with d(K,K^r) > g_4(K)

Abstract

The cobordism distance between knots, d(K,J), equals the four-genus g_4(K # -J). We consider d(K,K^r), where K^r is the reverse of K. It is elementary that 0 \le d(K,K^r) \le 2g_4(K) and it is known that there are knots K for which d(K,K^r) is arbitrarily large. Here it is shown that for any knot for which g_4(K) = g_3(K) (such as non-slice knots with g_3(K) = 1 or strongly quasi-positive knots), one has that d(K,K^r) is strictly less that twice g_4(K). It is shown that for arbitrary positive g, there exist knots for which d(K,K^r) = g = g_4(K). There are no known examples for which d(K,K^r) > g_4(K).