Twisted links and twisted virtual braids

Abstract

Twisted links correspond to stable equivalence classes of links in three-manifolds that are I-bundles over closed but not necessarily orientable surfaces. Virtual links, on the other hand, correspond to stable equivalence classes in I-bundles over oriented closed surfaces. Understanding these equivalence classes helps bridge classical knot theory with virtual and twisted knots. One major challenge in twisted knot theory is the less number of invariants that fully characterize these knots. We extend some invariants from virtual and classical knots, such as the odd writhe, arc shift number, and warping degree, can be adapted for twisted knots, offering new analytical tools. Another fundamental aspect of knot theory is the connection between knots and braids. Every classical link can be represented as the closure of a braid, and braid equivalence is governed by Markov moves. Similar theorems exist in virtual knot theory, and this work extends them to twisted links and twisted virtual braids. We introduce an invariant for twisted virtual braids based on the concept of warping degree, which has been studied in various settings, including classical, virtual, and welded knots. This invariant provides a new approach to analyzing twisted virtual braids, offering deeper insights into their structural properties. The twisted virtual braid group TV Bn is introduced, with its structure closely related to the classical braid group Bn. Various subgroups and epimorphisms of TV Bn onto Sn are studied, leading to decompositions that aid in understanding its structure. Additionally, we study singular twisted knots which are equivalence class of 4-valent graphs. These structures introduce twist-like features that differentiate them from conventional singular virtual knots. Inspired by existing results on Alexander- and Markov-like theorems for singular links and singular braids. We proved similar results for singular twisted links and singular twisted virtual braids. Also, proved that the set of singular twisted virtual braids on n strands forms a monoid.