Manifestations of Symmetry in Polynomial Link Invariants

Abstract

The use and detection of symmetry is ubiquitous throughout modern mathematics. In the realm of low-dimensional topology, symmetry plays an increasingly significant role due to the fact that many of the modern invariants being developed are computationally expensive to calculate. If information is known about the symmetries of a link, this can be incorporated to greatly reduce the computation time. This manuscript will consider graphical techniques that are amenable to such methods. First, we discuss an obstruction to links being periodic, developed jointly with Dr. Khaled Qazaqzeh at Kuwait University, using a model developed by Caprau and Tipton. We will discuss useful corollaries of this new method that arise when applying the criterion to multi-component links, and give a survey of its effectiveness when applied to low-crossing links. The second part will investigate a structure that arose in the model of Caprau and Tipton, namely singular links. We first define an invariant of singular links. We then develop a method based on the work of Turaev and Ohtsuki that allows for the creation of operator invariants from R-matrices. Finally, we show that the invariant defined previously is the natural extension of the Kauffman Bracket, when viewed through this framework. In the final section we investigate torus links, and relate the values of the Tail of the Colored Jones Polynomials of links within this family. This chapter involves well-known q-series, first studied by Ramanujan, and an unexpected combinatorial series related to planar integer partitions. This work was inspired by two seemingly unrelated questions of Robert Osburn and Oliver Dasbach. The first asked how the tails of two different links might be related, once one recognized that their Tait graphs have some shared structure. The latter asked if there might be a deletion/contraction type formulation for the Tail of the Colored Jones Polynomial, as it relates to the Tait graph of a link. This work is being done jointly with Mustafa Hajij at the University of South Florida.