A note on knot fertility

TL;DR

This paper proves that knots with an even minimum crossing number greater than 30 are not fertile, meaning they cannot produce certain other knots through simple crossing changes.

Contribution

It establishes a new class of non-fertile knots based on crossing number parity and size, expanding understanding of knot transformation limitations.

Findings

Knots with even crossing number >30 are not fertile.

Fertility depends on crossing number parity and size.

Certain knots cannot be obtained by crossing changes from minimal diagrams.

Abstract

We show that a knot whose minimum crossing number $c(K)$ is even and greater than $30$ is not fertile; there exists a knot $K'$ with crossing number less than $c$ such that $K'$ is not obtained from a minimum crossing number diagram of $K$ by suitably changing the over-under information.