Shadow movies not arising from knots

TL;DR

This paper demonstrates that not all sequences of shadow diagrams (shadow movies) correspond to actual sequences of classical knot diagrams (Reidemeister movies), impacting the interpretation of virtual crossings in knot theory.

Contribution

It proves that some shadow movies cannot be realized as classical Reidemeister movies, revealing limitations in the virtual knot framework.

Findings

Not all shadow movies arise from Reidemeister movies.

Virtual crossings cannot always be interpreted as classical crossings with undetermined over/under information.

Implications for the theory of virtual knots and their relation to classical knots.

Abstract

A shadow diagram is a knot diagram with under-over information omitted; a shadow movie is a sequence of shadow diagrams related by shadow Reidemeister moves. We show that not every shadow movie arises as the shadow of a Reidemeister movie, meaning a sequence of classical knot diagrams related by classical Reidemeister moves. This means that in Kaufman's theory of virtual knots, virtual crossings cannot simply be viewed as classical crossings where which strand is over has been left `to be determined'.