Prime component-preservingly amphicheiral link with odd minimal crossing number

TL;DR

This paper constructs prime component-preservingly amphicheiral links with any odd minimal crossing number at least 21, featuring a Stoimenow knot component that is non-invertible, expanding understanding of link symmetries.

Contribution

It introduces a new family of prime amphicheiral links with odd minimal crossing numbers, including non-invertible Stoimenow knots, for the first time.

Findings

Constructed prime amphicheiral links with odd crossing numbers ≥21

Demonstrated non-invertibility of Stoimenow knots via Alexander polynomials

Expanded the class of known amphicheiral links with specific crossing numbers

Abstract

For every odd integer $c\ge 21$, we raise an example of a prime component-preservingly amphicheiral link with the minimal crossing number $c$. The link has two components, and consists of an unknot and a knot which is $(-)$-amphicheiral with odd minimal crossing number. We call the latter knot a {\it Stoimenow knot}. We also show that the Stoimenow knot is not invertible by the Alexander polynomials.