The topological slice genus of satellite knots

Feller, Peter; Miller, Allison N; Pinzón-Caicedo, Juanita

doi:10.2140/agt.2022.22.709

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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

DOI: 10.2140/agt.2022.22.709

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 $ℤ$ –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 $n \in ℕ$ . 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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Keywords

4–genus, concordance, satellite knot, algebraic genus

Mathematical Subject Classification 2010

Primary: 57M25, 57N70

References

Publication

Received: 18 December 2019

Revised: 6 October 2020

Accepted: 14 November 2020

Published: 3 August 2022

Authors

Peter Feller
Department of Mathematics ETH Zurich Zurich Switzerland

Allison N Miller
Department of Mathematics and Statistics Swarthmore College Swarthmore, PA United States

Juanita Pinzón-Caicedo
Department of Mathematics University of Notre Dame Notre Dame, IN United States

Volume 22, issue 2 (2022)