On the slice genus of generalized algebraic knots

TL;DR

This paper constructs examples of algebraically slice knots with a topological and smooth four-genus of two, using Casson-Gordon invariants and cabling formulas to analyze their properties.

Contribution

It generalizes previous examples by providing a method to compute genus bounds for algebraically slice knots using Casson-Gordon invariants and cabling techniques.

Findings

Examples of algebraically slice knots with four-genus two.

Effective computation of Casson-Gordon invariants for these knots.

Extension of previous non-slice algebraically slice knot examples.

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.