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The topological slice
genus of satellite knots
Peter Feller, Allison N Miller and Juanita Pinzón-Caicedo |
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Algebraic & Geometric Topology 22 (2022) 709–738
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Abstract |
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We present evidence supporting the conjecture that, in the topological category, the slice genus of a satellite knot is bounded above by the sum of the slice genera of and . 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 . This stands in stark contrast with the smooth category, where, for example, there are many genus 1 knots whose –cables have arbitrarily large smooth –genera. As an application, we show that the –cable of any knot of –genus 1 (eg the figure-eight or trefoil knot) has topological slice genus at most 1, regardless of the value of . 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. |
Keywords
4–genus, concordance, satellite knot, algebraic genus
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Mathematical Subject Classification 2010
Primary: 57M25, 57N70
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References |
Publication
Received: 18 December 2019
Revised: 6 October 2020
Accepted: 14 November 2020
Published: 3 August 2022
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Authors |
| Peter Feller | |
| Department of Mathematics ETH Zurich Zurich Switzerland |
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| Allison N Miller | |
| Department of Mathematics and
Statistics Swarthmore College Swarthmore, PA United States |
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| Juanita Pinzón-Caicedo | |
| Department of Mathematics University of Notre Dame Notre Dame, IN United States |
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