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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

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 –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 . 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.

4–genus, concordance, satellite knot, algebraic genus
Mathematical Subject Classification 2010
Primary: 57M25, 57N70
Received: 18 December 2019
Revised: 6 October 2020
Accepted: 14 November 2020
Published: 3 August 2022
Peter Feller
Department of Mathematics
ETH Zurich
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