A note on knot fertility II

TL;DR

This paper introduces an obstruction criterion for knots to be (m,n)-fertile, explores the finiteness of such knots with specific parameters, and discusses related invariants like Seifert circles and writhe.

Contribution

It provides a new obstruction for (m,n)-fertility in knots and proves the finiteness of certain classes of fertile knots, advancing understanding of knot diagram transformations.

Findings

Obstruction criterion for (m,n)-fertility in knots

Finiteness of (c(K)+f,c(K)+p)-fertile knots for all f,p

Discussion on Seifert circles and writhe in minimal diagrams

Abstract

A knot $K$ is called $(m,n)$-fertile if for every prime knot $K'$ whose crossing number is less than or equal to $m$, there exists an $n$-crossing diagram of $K$ such that one can get $K'$ from the diagram by changing its over-under information. We give an obstruction for knot to be $(m,n)$-fertile. As application, we prove the finiteness of $(c(K)+f,c(K)+p)$-fertile knots for all $f,p$. We also discuss the nubmer of Seiefrt circle and writhe of minimum crossing diagrams.