The topological slice genus of satellite knots

Abstract

We present evidence supporting the conjecture that, in the topological category, the slice genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$. Our main result establishes this conjecture for a variant of the topological slice genus, the $\mathbb{Z}$–slice genus. Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern $P$. This stands in stark contrast with the smooth category, where, for example, there are many genus 1 knots whose $(n,1)$–cables have arbitrarily large smooth $4$–genera. As an application, we show that the $(n,1)$–cable of any knot of $3$–genus 1 (eg the figure-eight or trefoil knot) has topological slice genus at most 1, regardless of the value of $n\in \mathbb{N}$. Further, we show that the lower bounds on the slice genus coming from the Tristram–Levine and Casson–Gordon signatures cannot be used to disprove the conjecture.