Research Projects

Twisted link theory was discovered and formulated by M. O. Bourgoin in 2008 as a generalization of virtual knot theory. Twisted links correspond to stable equivalence classes of links in 3-manifolds that are oriented bundles over closed but not necessarily orientable surfaces. A twisted link is an embedding of finitely many disjoint circles into the oriented 3-manifold Sg × I, where Sg is a surface of genus g ( possibly non-orientable). A twisted knot is a particular case of twisted links when we consider embedding one circle in an oriented 3-manifold. Twisted link theory is the study of twisted links considered equivalent to homeomorphisms of the surface Sg, isotopies of the ambient space Sg × I, and the addition and removal of handles or Mobius bands from Sg ×I disjoint from the embedded closed curves. Likewise, knot theory has a diagrammatic description in which the embedded circles are encoded as regular diagrams in the plane, and extended Reidemeister moves imitate the notion of equivalence. A twisted link diagram is one in the plane with real or virtual crossings and possibly with bars representing twists. A virtual crossing at a double point is conventionally denoted by putting a small circle around it. A twisted link is the equivalence class of a twisted link diagram with respect to the equivalence relation generated by certain moves called extended Reidemeister moves. This diagram-based approach to studying the topological properties of twisted links is simple and predominant in theory. However, given any two twisted link diagrams, it is generally difficult to decide if they represent the same twisted link or not. One usually associates a twisted link diagram with a quantity that does not change under the extended set of Reidemeister moves to distinguish twisted links from each other. Such a quantity, or a property, is called an invariant of the twisted link. Then the values of such invariants are compared to tell things apart, which, at times, is inconclusive if the values coincide. Twisted virtual braids form another set of topological objects that interest knot theorists. Recently, we gave the notion of twisted virtual braids and established Alexander’s and Markov’s Theorems, which show a strong relation between twisted virtual braids and twisted links (we uploaded this work in the archive recently). Alexander’s theorem states that every twisted link is represented as the closure of a twisted virtual braid. Markov’s theorem states that such a twisted virtual braid is unique modulo certain moves, called Markov moves. In this research project, we want to define, examine, and utilize new invariants of twisted links and virtual braids. We want to examine the topological and geometrical properties of twisted links and virtual braids, which can be achieved by studying the combinatorial data associated with their planar diagrams.