[Paper Review] A note on the topological slice genus of satellite knots

Peter Feller, Allison N. Miller+1 more|arXiv (Cornell University)|Aug 10, 2019

Geometric and Algebraic Topology23 references4 citations

TL;DR

This paper investigates the topological slice genus of satellite knots, proposing that the genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of the companion knot $K$ and the pattern knot $P(U)$. The authors prove this conjecture for the $\mathbb{Z}$-slice genus variant and apply it to show that the $(n,1)$-cable of any 3-genus 1 knot has topological slice genus at most 1, highlighting a key contrast with the smooth category where such genera can be arbitrarily large.

ABSTRACT

This paper presents evidence supporting the surprising 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 $\mathbb{Z}$-slice genus. As an application, we show that the $(n,1)$-cable of any 3-genus 1 knot (e.g. the figure 8 or trefoil knot) has topological slice genus at most 1. 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. 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.

Motivation & Objective

• To investigate the behavior of the topological slice genus under satellite operations.
• To test the conjecture that the topological slice genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$.
• To explore whether classical invariants like Tristram-Levine and Casson-Gordon signatures can disprove the conjecture.
• To establish the conjecture for the $\mathbb{Z}$-slice genus, a variant of the topological slice genus.
• To demonstrate that the conjectured bound does not depend on the algebraic winding number of the pattern $P$, contrasting with the smooth category.

Proposed method

• The authors introduce and analyze the $\mathbb{Z}$-slice genus, a variant of the topological slice genus that restricts to integer homology cobordism.
• They use concordance invariants and cobordism techniques to bound the $\mathbb{Z}$-slice genus of satellite knots.
• The proof relies on constructing explicit topological concordances via the satellite construction, leveraging the structure of the pattern and companion knots.
• They apply the $\mathbb{Z}$-slice genus to analyze $(n,1)$-cables of 3-genus 1 knots, such as the trefoil and figure-eight.
• They show that classical signature invariants—Tristram-Levine and Casson-Gordon—cannot obstruct the conjectured upper bound.
• The argument avoids dependence on the algebraic winding number of the pattern $P$, distinguishing the topological from the smooth category.

Experimental results

Research questions

• RQ1 Can the topological slice genus of a satellite knot $P(K)$ be bounded above by the sum of the slice genera of $K$ and $P(U)$?
• RQ2 Does the $\mathbb{Z}$-slice genus satisfy the same upper bound as the standard topological slice genus for satellite knots?
• RQ3 Can classical signature invariants such as Tristram-Levine and Casson-Gordon signatures be used to disprove the conjectured bound?
• RQ4 Why does the conjectured bound not involve the algebraic winding number of the pattern $P$, unlike in the smooth category?
• RQ5 What is the topological slice genus of the $(n,1)$-cable of a 3-genus 1 knot, such as the trefoil or figure-eight?

Key findings

• The conjectured upper bound on the topological slice genus of satellite knots holds for the $\mathbb{Z}$-slice genus, providing a partial proof of the main conjecture.
• The $(n,1)$-cable of any 3-genus 1 knot has topological slice genus at most 1, as a consequence of the $\mathbb{Z}$-slice genus result.
• Classical signature invariants such as Tristram-Levine and Casson-Gordon signatures cannot be used to disprove the conjectured upper bound.
• The conjectured bound does not depend on the algebraic winding number of the pattern $P$, which contrasts sharply with the smooth category where such dependence leads to arbitrarily large genera.
• The result demonstrates a fundamental difference between the topological and smooth categories: in the topological setting, satellite operations do not increase the slice genus beyond the sum of the components' genera.