Document Type

Article

Publication Date

2022

Published In

Algebraic And Geometric Topology

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∈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.

Keywords

4–genus, concordance, satellite knot, algebraic genus

Comments

This work is a preprint that is freely available from arXiv.org at arXiv:1908.03760, courtesy of Mathematical Sciences Publishers.

Download

Included in

Mathematics Commons

Share

COinS