Doctor of Philosophy (PhD)
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 Krebes’ 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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Abernathy, Susan Marie, "Obstructions to Embedding Genus-1 Tangles in Links" (2014). LSU Doctoral Dissertations. 3251.