There exist knots that have positive and negative 4-dimensional clasp numbers zero but have 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 are of infinite order in the concordance group, rather than being two-torsion. An added feature of the examples here is their simplicity; all are two-bridge knots and include the two-bridge knot B(25,2), the first algebraically slice knot that was proved to be non-slice by Casson and Gordon in 1973.

This work was supported by a grant from the National Science Foundation, NSF-DMS-1505586.