A note on the four-dimensional clasp number of knots

TL;DR

This paper characterizes certain knots with specific 4-dimensional clasp numbers and demonstrates how their properties scale under connected sums, contrasting topological and smooth categories.

Contribution

It provides a characterization of knots with clasp number at least 2 and shows how these properties extend to their connected sums, including a construction of topologically slice knots with similar behavior.

Findings

Knots with cobordism distance 1 can have 4-dimensional clasp number at least 2.

The 4-dimensional clasp number scales linearly with connected sums.

Construction of topologically slice knots with high clasp number growth.

Abstract

Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterize those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number at least 2n. Our proof works in the topological category. To contrast this, we build a family of topologically slice knots for which the n-fold connected self-sum has 4-ball genus n and 4-dimensional clasp number at least 2n.