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Full-Text Articles in Physical Sciences and Mathematics
On Matroid And Polymatroid Connectivity, Dennis Wayne Hall Ii
On Matroid And Polymatroid Connectivity, Dennis Wayne Hall Ii
LSU Doctoral Dissertations
Matroids were introduced in 1935 by Hassler Whitney to provide a way to abstractly capture the dependence properties common to graphs and matrices. One important class of matroids arises by taking as objects some finite collection of one-dimensional subspaces of a vector space. If, instead, one takes as objects some finite collection of subspaces of dimensions at most k in a vector space, one gets an example of a k-polymatroid.
Connectivity is a pivotal topic of study in the endeavor to understand the structure of matroids and polymatroids. In this dissertation, we study the notion of connectivity from several …
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 …