Parities for virtual braids and string links

Abstract

Virtual knot theory is an extension of classical knot theory based on a combinatorial presentation of crossing information. The appropriate extensions of braid groups and string link monoids have also been studied. While some previously known knot invariants can be evaluated for virtual objects, entirely new techniques can also be used, for example, the concept of index of a crossing, and its resulting (Gaussian) parity theory. In general, a parity is a rule which assigns 0 or 1 to each crossing in a knot or link diagram. Recently, they have also been defined for virtual braids. Here, novel parities for knots, braids, and string links are defined, some of their applications are explored, most notably, defining a new subgroup of the virtual braid groups.