#### Volume 22, issue 2 (2022)

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The topological slice genus of satellite knots

### Peter Feller, Allison N Miller and Juanita Pinzón-Caicedo

Algebraic & Geometric Topology 22 (2022) 709–738
##### Abstract

We present evidence supporting the conjecture that, in the topological category, the slice genus of a satellite knot $P\left(K\right)$ is bounded above by the sum of the slice genera of $K$ and $P\left(U\right)$. Our main result establishes this conjecture for a variant of the topological slice genus, the $ℤ$–slice genus. Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern $P\phantom{\rule{-0.17em}{0ex}}$. This stands in stark contrast with the smooth category, where, for example, there are many genus 1 knots whose $\left(n,1\right)$–cables have arbitrarily large smooth $4$–genera. As an application, we show that the $\left(n,1\right)$–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 $n\in ℕ$. 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.

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