Crosscap number and knot projections

TL;DR

This paper introduces a new unknotting-type number for knot projections that bounds the crosscap number of knots, and characterizes projections with low unknotting-type numbers, impacting the understanding of alternating knots.

Contribution

It defines a novel unknotting-type number for knot projections and characterizes those with low values, linking to classical and new results on crosscap numbers.

Findings

Set of knot projections with unknotting-type number ≤ 2 determined

Implication for classifying alternating knots with crosscap number ≤ 2

Provides bounds on crosscap number using a new projection-based invariant

Abstract

We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical and new results that determine the set of alternating knots with the crosscap number at most two.