Doctor of Philosophy (PhD)
In this work we consider the Z1(K) polynomial time knot polynomial defined and
described by Dror Bar-Natan and Roland van der Veen in their 2018 paper ”A polynomial time knot polynomial”. We first look at some of the basic properties of Z1(K), and develop an invariant of diagrams Ψm(D) related to this polynomial. We use this invariant as a model to prove how Z1(K) acts under the connected sum operation. We then discuss the effect of mirroring the knot on Z1(K), and described a geometric interpretation of some of the building blocks of the invariant. We then use these to develop state sum interpretation of Z1(K). We describe a base set of knots which can be used to build the Z1(K), or rather its normalization ρ1(K), showcasing some of its symmetry properties. Finally, we use this idea to give an explicit expansion of ρ1(K) for the family of T (2, 2p + 1) torus knots in terms of this base set of knot invariants.
Quarles, Robert John, "A New Perspective on a Polynomial Time Knot Polynomial" (2022). LSU Doctoral Dissertations. 5806.