Title page for ETD etd-07042014-141943


Type of Document Dissertation
Author Abernathy, Susan Marie
Author's Email Address susan.abernathy@gmail.com
URN etd-07042014-141943
Title Obstructions to Embedding Genus-1 Tangles in Links
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Gilmer, Patrick Committee Chair
Achar, Pramod Committee Member
Dani, Pallavi Committee Member
Litherland, Richard Committee Member
Oxley, James Committee Member
Peck, Robert Dean's Representative
Keywords
  • tangle embedding
  • tangles
  • knot theory
  • low-dimensional topology
Date of Defense 2014-05-01
Availability unrestricted
Abstract
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 G? This question is motivated by a particular example of a genus-1 tangle given by Krebes, which we denote by A.

Krebes asks whether the unknot is a closure of A. We partially answer this question in Chapter 1 using a theorem of Ruberman and cyclic branched covers of the solid torus branched over A. We prove that if Krebesí tangle A embeds in the unknot, then A must be completed to the unknot by an arc which passes through the hole of the solid torus containing A an even number of times.

In Chapter 3, we discuss the Kauffman bracket ideal, which gives an obstruction to tangle embedding for general (M,2n)-tangles. For each tangle T in M, we define an ideal I_T called the Kauffman bracket ideal. It is easy to see that if I_T is non-trivial, then T does not embed in the unknot. Using skein theory, we give an algorithm for computing a finite list of generators for the Kauffman bracket ideal of any genus-1 tangle, and give an example of a genus-1 tangle with non-trivial Kauffman bracket ideal. We also explore the relationship between partial closures of tangles and this ideal.

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