The Number of Vogel Operations to Deform a Link Diagram to a Closed Braid

Abstract

Vogel showed that any oriented link diagram $D$ can be deformed to a closed braid by a finite sequence of Reidemeister II moves, each performed on two coherently oriented edges in a face of $D$ such that the edges are contained in distinct Seifert circles. We show that the number of such moves is constant for a given oriented link diagram, and does not depend on the sequence of moves. An easy way of calculating the number is given.