On Braids, Branched Covers and Transverse Invariants

Abstract

In this work, we present a brief survey of knot theory supported by contact 3-manifolds. We focus on transverse knots and explore different ways of studying transverse knots. We define a new family of transverse invariants, this is accomplished by considering $n$-fold cyclic branched covers branched along a transverse knot and we then extend the definition of the BRAID invariant $t$ defined in cite{BVV} to the lift of the transverse knot. We call the new invariant the lift of the BRAID invariant and denote it by $t_n$. We then go on to show that $t_n$ satisfies a comultiplication formula and use this result to prove a vanishing theorem for $t_n$. We also re-prove a previously known result regarding the $n$-fold branched covers branched along stabilized transverse knot. We use this result to prove another vanishing result for $t_n$.