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Home CCSU Theses & Dissertations Minimal Embeddings of Knots in the Cubic Lattice

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Minimal Embeddings of Knots in the Cubic Lattice

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Identifier	Thesis 1984
Author	Wysong, Kimberly Ann, 1979-
Title	Minimal Embeddings of Knots in the Cubic Lattice
Publisher	Central Connecticut State University
Date	2008
Resource Type	Master's Thesis
Notes	steps required to represent the knot as a polygon in the cubic lattice. Several lower bounds for the lattice step numbers of different knots have been obtained by computer simulations but the exact value of the lattice step number seems to be known only for the trefoil knot which is the simplest nontrivial knot. In this thesis we analyze three different characteristics of minimal embeddings in the cubic lattice. In Chapter 2, we study the work of Diao that proves that the minimum number of steps required to produce a nontrivial knot in the cubic lattice is 24. In Chapter 3, we study two works. First, the proof of van Rensburg and Promislov regarding the curvature of lattice knots which is essentially the number of turns of the polygon. For their investigations, these authors develop a technique that uses weighted sequences and functions. We elaborate on their theorems by providing examples and diagrams that we hope help to clarify their work. Second, we look at the proof of Huh and Oh who develop methods to study the minimal stick number of the trefoil and the gure eight knots. In Chapter 4, we use the techniques of Diao to investigate the lattice step number of the gure eight knot. We prove that several of Diao's results can be adapted and applied to the four crossing knot. Our results support the conjecture that the lattice step number of the gure eight knot is 30.
Subject	Knot theory
Department	Department of Mathematical Sciences
Advisor	Castañeda, Nelson
Type	Text
Digital Format	application/pdf
Language	eng
OCLC number	713734310
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Minimal Embeddings of Knots in the Cubic Lattice

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