On the slice genus of generalized algebraic knots

Abstract

We give examples of a linear combination of algebraic knots and their mirrors that are algebraically slice, but whose topological and smooth four-genus is two. Our examples generalize an example of non-slice algebraically slice linear combination of iterated torus knots obtained by Hedden, Kirk and Livingston. Our main tool is a genus bound from Casson--Gordon invariants and a cabling formula that allows us to compute effectively these invariants.

6 pages, 1 figure. In v2 we did some minor corrections. In v3 the paper underwent a substantial revision. Hopefully, it should be more accessible to non-experts