DOI

10.17077/etd.8xz610tq

Document Type

Dissertation

Date of Degree

Summer 2016

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Frohman, Charles

First Committee Member

Cooper, Ben

Second Committee Member

Tomova, Maggy

Third Committee Member

Strohmer, Gerhard

Fourth Committee Member

Durumeric, Oguz

Abstract

This thesis has four chapters. After a brief introduction in Chapter 1, the $AJ$-conjecture is introduced in Chapter 2. The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If $K$ satisfies the $AJ$-conjecture, sufficient conditions on $K$ are given for the $(r,2)$-cable knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram of $K$ has $\eta_+$ positive crossings and $\eta_-$ negative crossings, then $C$ will satisfy the $AJ$-conjecture when $(r+4\eta_-)(r-4\eta_+)>0$ and the conditions of Theorem 2.2.1 are satisfied. Chapter 3 is about quantum curves and their relation to the $AJ$ conjecture. The variables $l$ and $m$ of the $A$-polynomial are quantized to operators that act on holomorphic functions. Motivated by a heuristic definition of the Jones polynomial from quantum physics, an annihilator of the Chern-Simons section of the Chern-Simons line bundle is found. For torus knots, it is shown that the annihilator matches with that of the colored Jones polynomial. In Chapter 4, a tangle functor is defined using semicyclic representations of the quantum group $U_q(sl_2)$. The semicyclic representations are deformations of the standard representation used to define Kashaev's invariant for a knot $K$ in $S^3$. It is shown that at certain roots of unity the semicyclic tangle functor recovers Kashaev's invariant.

Keywords

AJ Conjecture, Chern-Simons, Colored Jones Polynomial, Quantum, Semicyclic, Tangle Functors

Pages

vi, 65 pages

Bibliography

Includes bibliographical references (pages 61-65).

Copyright

Copyright © 2016 Nathan Druivenga

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