Knot fertility and lineage

Abstract

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot [Formula: see text] is a descendant of another knot [Formula: see text] if [Formula: see text] can be obtained from a minimal crossing diagram of [Formula: see text] by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration.