A note on knot fertility II

Abstract

A knot $K$ is called $(m,n)$-fertile if for every prime knot $K^{\prime }$ whose crossing number is less than or equal to $m$, there exists an $n$-crossing diagram of $K$ such that one can get $K^{\prime }$ 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.

7 pages, no Figures

References in corpus (2)

• A quantitative Birman-Menasco finiteness theorem and its application to crossing number
• A note on knot fertility