Files and links (1)
Free to read and download, Open Access
Dissertation
Quantum topology and me
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2016
DOI: 10.17077/etd.8xz610tq
Abstract
<p>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.</p>
Details
- Title: Subtitle
- Quantum topology and me
- Creators
- Nathan Druivenga - University of Iowa
- Contributors
- Charles Frohman (Advisor)Ben Cooper (Committee Member)Maggy Tomova (Committee Member)Gerhard Strohmer (Committee Member)Oguz Durumeric (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Summer 2016
- Publisher
- University of Iowa
- DOI
- 10.17077/etd.8xz610tq
- Number of pages
- vi, 65 pages
- Copyright
- Copyright © 2016 Nathan Druivenga
- Language
- English
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (pages 61-65).
- Academic Unit
- Mathematics
- Record Identifier
- 9983777299402771
Metrics
357 File views/ downloads
498 Record Views