On fertility of knot shadows

Abstract

A knot [Formula: see text] is a parent of a knot [Formula: see text] if there exists a minimal crossing diagram [Formula: see text] of [Formula: see text] such that a subset of the crossings of [Formula: see text] can be changed to produce a diagram of [Formula: see text]. A knot [Formula: see text] with crossing number [Formula: see text] is fertile if for any prime knot [Formula: see text] with crossing number less than [Formula: see text], [Formula: see text] is a parent of [Formula: see text]. It is known that only [Formula: see text] are fertile for knots up to 10 crossings. However it is unknown whether there exist other fertile knots. A knot shadow is a diagram without over/under information at all crossings. In this paper, we introduce a definition of fertility for knot shadows. We show that if an alternating knot [Formula: see text] is fertile then the crossing number of [Formula: see text] is less than eight.