Danceability, A New Definition of Bridge Index

Abstract

There are many commonly known definitions of the bridge index coming from combinatorial knot theory, Morse theory, geometry, and algebra. In this paper, we prove that the danceability index is a new equivalent definition of the bridge index which can be seen as an oriented version of the Wirtinger number. 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.