Extending Van Cott's bounds for the $\tau$-invariant of satellite knots

TL;DR

This paper extends Van Cott's bounds on the $ au$-invariant from cabled knots to all satellite knots using geometric methods, broadening the understanding of knot invariants without Heegaard-Floer homology.

Contribution

It generalizes existing bounds on the $ au$-invariant to a wider class of satellite knots through geometric techniques, avoiding Heegaard-Floer homology.

Findings

Extended bounds apply to all satellite knots

Achieved bounds without using Heegaard-Floer homology

Provided new geometric methods for knot invariant analysis

Abstract

We generalize C. Van Cott's results on the $\tau$ and $s$-invariants of cabled knots to apply to general satellite knots. This paper uses no Heegaard-Floer homology, relying on more geometric techniques.