A method of encoding generalized link diagrams

Abstract

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of 'riser marks'. For classical knots, the minimal number of such marks is twice the bridge index, and a classical knot diagram in minimal bridge form with bridge index $b$ may be encoded in space $\mathcal{O}(b^2)$. A set of moves on the notation is defined. As a demonstration of the utility of the notation we give another proof that the Kishino virtual knot is non-classical.