Danceability, A New Definition of Bridge Index

TL;DR

This paper introduces the danceability index as a new, equivalent definition of the bridge index in knot theory, extending it to virtual knots and comparing it with existing notions.

Contribution

It establishes danceability as a new bridge index definition and extends it to virtual knots, providing a novel perspective in knot invariants.

Findings

Danceability index is equivalent to the bridge index.

Extended danceability to virtual knots in multiple ways.

Compared new invariants with existing virtual knot bridge indices.

Abstract

There are three commonly known definitions of the bridge index. These definitions come from combinatorial knot theory, Morse theory, and geometry. In this paper, we prove that the danceability index is a fourth equivalent definition of the bridge index. We extend the danceability invariant to virtual knots in multiple ways and compare these invariants to two different notions of bridge index for virtual knots.