Signed clasp numbers and four-genus bounds

TL;DR

This paper explores knots with zero positive and negative four-dimensional clasp numbers but arbitrarily large four-genus, including infinite order examples, expanding understanding of knot concordance and clasp number relationships.

Contribution

It constructs new examples of knots with specific clasp number properties, including infinite order in concordance, extending previous work by Miller.

Findings

Knots with zero clasp numbers can have arbitrarily large four-genus.

New infinite order examples in concordance are provided.

The results expand the understanding of the relationship between clasp numbers and four-genus.

Abstract

There exist knots having positive and negative four-dimensional clasp numbers zero but having four-genus, and hence clasp number, arbitrarily large. Such examples were first constructed by Allison Miller, answering a question of Juhasz-Zemke. Further examples are constructed here, complementing those of Miller in that they include examples that are of infinite order in concordance.