The Hyperbolic Structure of the Complements of Rational Links with Conway Notation mn for m,n≥2

File:Surmathesis.pdfShow FileMIME type:application/pdfFile Size:1.8 Mb
Surmathesis.pdf
Abstract
We use a specific application of Adams algorithm to investigate the hyper
bolic structure of rational links composed of two tangles. The investigation
begins with the gureeight knot (the simplest hyperbolic knot), and we note
changes in the hyperbolic structure as halftwists are added to each of its com
ponent tangles. We model the complements of the links under investigation in
a speci c way such that the following patterns are observed. Our model for
the complement of a link with Conway notation mn, m, n ≥ 2, will contain
(m + n)  3 pairs of tetrahedra and 2((m + n)  3) edge types. Furthermore,
the tetrahedral con guration and edge equations corresponding to these models
change in a predictable way as we each additional halftwist. The 2((m+ n)3)
edge equations are then simpli ed down to (m + n) 3 equations and solved.
Their solutions determine the hyperbolic structure and are used to determine
the hyperbolic volume of each knot under investigation. As m and n increase,
the growth rate of the hyperbolic volume decreases.
Description
Franklin and Marshall College Archives, Undergraduate Honors Thesis 2011
Collections
 F&M Theses Collection [291]
Date
2011Author
Surma, Mark
Metadata
Show full item recordAuthor  Surma, Mark  
Date Accessioned  20110527T18:36:00Z  
Date Available  20110527T18:36:00Z  
Date of Issue  2011  
Identifier (URI)  http://hdl.handle.net/11016/22105  
Description  Franklin and Marshall College Archives, Undergraduate Honors Thesis 2011  en_US 
Description  We use a specific application of Adams algorithm to investigate the hyper bolic structure of rational links composed of two tangles. The investigation begins with the gureeight knot (the simplest hyperbolic knot), and we note changes in the hyperbolic structure as halftwists are added to each of its com ponent tangles. We model the complements of the links under investigation in a speci c way such that the following patterns are observed. Our model for the complement of a link with Conway notation mn, m, n ≥ 2, will contain (m + n)  3 pairs of tetrahedra and 2((m + n)  3) edge types. Furthermore, the tetrahedral con guration and edge equations corresponding to these models change in a predictable way as we each additional halftwist. The 2((m+ n)3) edge equations are then simpli ed down to (m + n) 3 equations and solved. Their solutions determine the hyperbolic structure and are used to determine the hyperbolic volume of each knot under investigation. As m and n increase, the growth rate of the hyperbolic volume decreases.  en_US 
Sponsorship  Franklin and Marshall College, Department of Mathematics  en_US 
Language  en_US  en_US 
Subject  Hyperbolic 3Manifolds, mathematics  en_US 
Subject  Mathematics  
Title  The Hyperbolic Structure of the Complements of Rational Links with Conway Notation mn for m,n≥2  en_US 
Type  Thesis  en_US 
Related items
Showing items related by title, author, creator and subject.

Hyperbolic 3Manifolds: Dehn Surgery Approach to the FigureEight Knot Complement
Dubicki, Sarah (2011)This project provides an examination of knot complements and their relationship to hyperbolic 3manifolds. It begins with the study of knot theory, specifically knot complements in S. The knots we will be focusing on are ... 
Drawing [Design] of a castle with title "Square Root", dated February 7, 1828
P. W. W. (1828)Drawing of a castle with title "Square Root", dated February 7, 1828. Signed "by P.W.W." Contains mathematical calculations. From a scrapbook of drawings by unknown artist. 
QSphere
Robbin (unknown), TonyMultifaceted sculpture made of metal and colored Plexiglas in a circular arrangement