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 \mathcal {C} . There are examples of knots for which K ≠ K r ∈ C K \ne K^r \in \mathcal {C} . This paper studies the cobordism distance, d ( K , K r ) d(K, K^r) . If K ≠ K r ∈ C K \ne K^r \in \mathcal {C} , then d ( K , K r ) > 0 d(K, K^r) >0 and it is elementary to see that for all K K , d ( K , K r ) ≤ 2 g 4 ( K ) d(K, K^r) \le 2g_4(K) , where g 4 ( K ) g_4(K) denotes the four-genus. Here we present a proof that for non-slice knots satisfying g 3 ( K ) = g 4 ( K ) g_3(K) = g_4(K) , one has d ( K , K r ) ≤ 2 g 4 ( K ) − 1 d(K,K^r) \le 2g_4(K) -1 . This family includes all strongly quasi-positive knots and all non-slice genus one knots. We also construct knots K K of arbitrary four-genus for which d ( K , K r ) = g 4 ( K ) d(K,K^r) = g_4(K) . Finding knots for which d ( K , K r ) > g 4 ( K ) d(K,K^r) > g_4(K) remains an open problem.

Knot reversal and rational concordance

“…Recently, Livingston and the author [11] showed that there are topologically slice knots that are not concordant to their reverses. Livingston [13] also showed relationships between a knot and its reverse in terms of cobordism distance between knots.…”