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Manifestations Of Symmetry In Polynomial Link Invariants, Kyle Istvan
Manifestations Of Symmetry In Polynomial Link Invariants, Kyle Istvan
LSU Doctoral Dissertations
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 …
On The Skein Theory Of 0-Framed Surgery Along The Trefoil Knot, Andrew Robert Holmes
On The Skein Theory Of 0-Framed Surgery Along The Trefoil Knot, Andrew Robert Holmes
LSU Doctoral Dissertations
In this dissertation, we will give a generating set of the Kauffman bracket skein module over the field Q(A) of 0-framed surgery along the trefoil knot. This generating set is described as a certain subset of a known basis for the skein module over Z[A^±1] of the trefoil exterior.
Beyond The Tails Of The Colored Jones Polynomial, Jun Peng
Beyond The Tails Of The Colored Jones Polynomial, Jun Peng
LSU Doctoral Dissertations
In [2] Armond showed that the heads and tails of the colored Jones polynomial exist for adequate links. This was also shown independently by Garoufalidis and Le for alternating links in [8]. Here we study coefficients of the "difference quotient" of the colored Jones polynomial. We begin with the fundamentals of knot theory. A brief introduction to skein theory is also included to illustrate those necessary tools. In Chapter 3 we give an explicit expression for the first coefficient of the relative difference. In Chapter 4 we develop a formula of t_2, the number of regions with exactly 2 crossings …
Obstructions To Embedding Genus-1 Tangles In Links, Susan Marie Abernathy
Obstructions To Embedding Genus-1 Tangles In Links, Susan Marie Abernathy
LSU Doctoral Dissertations
Given a compact, oriented 3-manifold M in S3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S3 if T can be completed to L by adding a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We focus on the case of (S_1 x D_2, 2)-tangles, also called genus-1 tangles, and consider the following question: given a genus-1 tangle G and a link L, how can we tell if L is a closure of …
Skein Theory And Topological Quantum Field Theory, Xuanting Cai
Skein Theory And Topological Quantum Field Theory, Xuanting Cai
LSU Doctoral Dissertations
Skein modules arise naturally when mathematicians try to generalize the Jones polynomial of knots. In the first part of this work, we study properties of skein modules. The Temperley-Lieb algebra and some of its generalizations are skein modules. We construct a bases for these skein modules. With this basis, we are able to compute some gram determinants of bilinear forms on these skein modules. Also we use this basis to prove that the Mahler measures of colored Jones polynomial of a sequence of knots converges to the Mahler measure of some two variable polynomial. The topological quantum field theory constructed …
The Head And Tail Conjecture For Alternating Knots, Cody Armond
The Head And Tail Conjecture For Alternating Knots, Cody Armond
LSU Doctoral Dissertations
The colored Jones polynomial is an invariant of knots and links, which produces a sequence of Laurent polynomials. In this work, we study new power series link invariants, derived from the colored Jones polynomial, called its head and tail. We begin with a brief survey of knot theory and the colored Jones polynomial in particular. In Chapter 3, we use skein theory to prove that for adequate links, the n-th leading coefficient of the N-th colored Jones polynomial stabilizes when viewed as a sequence in N. This property allows us to define the head and tail for adequate links. In …
Dimer Models For Knot Polynomials, Moshe Cohen
Dimer Models For Knot Polynomials, Moshe Cohen
LSU Doctoral Dissertations
A dimer model consists of all perfect matchings on a (bipartite) weighted signed graph, where the product of the signed weights of each perfect matching is summed to obtain an invariant. In this paper, the construction of such a graph from a knot diagram is given to obtain the Alexander polynomial. This is further extended to a more complicated graph to obtain the twisted Alexander polynomial, which involved "twisting" by a representation. The space of all representations of a given knot complement into the general linear group of a fixed size can be described by the same graph. This work …