This dissertation concerns the study of canonical components of the SL(2, C) character varieties of hyperbolic 3-manifolds. Although character varieties have proven to be a useful tool in studying hyperbolic 3-manifolds, very little is known about their structure. Chapter 1 provides background on this subject. Chapter 2 is dedicated to the canonical component of the Whitehead link. We provide a projective model and show that this model is isomorphic to P^2 blown up at 10 points. The Whitehead link can be realized as 1/1 Dehn surgery on one cusp of both the Borromean rings and the 3-chain link. In Chapter 3 we examine the canonical components for the two families of hyperbolic link complements obtained by 1/n Dehn filling on one component of both the Borromean rings and the 3-chain link. These examples extend the work of Macasieb, Petersen and van Luijk who have studied the character varieties associated to the twist knot complements. We conjecture that the canonical components for the links obtained by 1/n Dehn filling on one component of the 3-chain link are all rational surfaces isomorphic to P^2 blown up at 9n + 1 points. A major goal is to understand how the algebro-geometric structure of these varieties reflects the topological structure of the associated manifolds. At the end of Chapter 3 we discuss common features of these examples and explain how our results lend insight into the affect Dehn surgery has on the character variety. We conclude, in Chapter 4, with a description of possible directions for future research. / text

International Telemetering Conference Proceedings / October 21, 2002 / Town & Country Hotel and Conference Center, San Diego, California / Using a network-based telemetry system places additional requirements on the Radio Frequency (RF) link. Limitations imposed by this link must be considered in advance when designing a network-based telemetry system.

International Telemetering Conference Proceedings / October 21, 2002 / Town & Country Hotel and Conference Center, San Diego, California / A three dimension (3D) link analysis is performed considering multipath effects caused by a reflected signal and the difference angle between the antenna bore-sight and Line-Of-Sight (LOS). In addition, a direction of a receive antenna is determined for a receiver to get maximum signal strength in a telemetry situation. For a fixed receive antenna, the angle is determined to maximize the average Carrier to Noise Ratio (CNR) over the interested part of a trajectory. For a tracking antenna, the angle at every position is selected to give maximum CNR or to direct the boresight to the flying projectile.

In this work a new foundation for the study of positive braids in Artin's braid groups is given. The basic braids considered are the set SBn of positive permutation braids, defined as those positive braids where each pair of arcs cross at most once. These are shown to be in 1-1 correspondence with the permutations in S . A canonical n form for positive braids as products of braids in SB is given, ton gether with an explicit algorithm for writing every positive braid in canonical form and a practical test for use in the algorithm. This is a useful approach to braid theory because permutations can be particularly easily handled. Applications of this canonical form are: (1) An easily handled approach to Garside's solution of the word problem in B . n (2) An algorithm to decide whether (/1 ) k is a factor of a positive n braid; this happens if and only if at most k canonical factors have equal to /1 n (where /1 n is the positive braid with each pair of arcs cross exactly once). (3) A proof that a positive braid is a factor of (/1 ) k if and only if n its canonical form has at most k factors. (4) An improvement of Garside's solution of the conjugacy problem, this is by reducing the summit set to a much smaller invariant class under conjugation (super summit set). This includes a necessary and sufficient condition for positive braid to contain /1 n up to conjugation. The linear generators of the Hecke algebras used by Morton. and/ Short are in 1-1 correspondence with the elements of SB. The n canonical forms above give a quick proof that the number of strands in a twist positive braid (one of the form (/1 )2mp for positive braid n P and for positive integer m) is the braid index of the closure of that braid, which was first proved by Franks and Williams. It is also shown that if the 2-variable link invariant P L (v, z) for an oriented link L has width k in the variable v, then it is the same as the polynomial of a closed k-braid, for k = 1, 2. A complete list of 3-braids of width 2, which close to knots, is given. It is also shown that twist positive 3-braids do not admit exchange moves (in the sense of Birman). Consequently the conjugacy class of a twist positive 3-braid representative is a complete link invariant, provided that Birman's conjecture about Markov's moves and exchange moves holds. Lorenz knots and links are studied as an example of positive links. It is proved that a positive permutation braid 1T is a Lorenz braid if and only if all braid words which equal 1T have the same single starting letter. A semicanonical form for a minimal braid representative of a Lorenz link is given. It is proved that every algebraic link of c components is a Lorenz link, for c = 1, 2. (The case for knots was first proved by Birman and Williams). Consequently a necessary and sufficient condition for a knot to be algebraic is given, together with a canonical form for a minimal braid representative for every algebraic knot. To some extent the relation between Lorenz knots and their companions is studied. It is shown that Lorenz knots and links of braid index 3 are determined by conjugacy classes in B 3. A complete list of 3 -braids which close to Lorenz knots and links is given and a complete list of pure 4-braids which close to Lorenz links is also given. It is shown that Lorenz knots and links of braid index 3 are determined by their Alexander polynomials. As a further analogy with the properties of algebraic links it is shown that the linking pattern of a Lorenz link L with pure braid representative and braid index t ~4, determines a unique braid representative for L and so determines L.

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Cheung, Chun Ngai. / "August 2011." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 55-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Coding the Lorenz knots --- p.10 / Chapter 2.1 --- Lorenz words --- p.10 / Chapter 3 --- Lorenz links and positive braids --- p.15 / Chapter 3.1 --- Lorenz braids --- p.15 / Chapter 3.2 --- Properties of Lorenz links as the closure of a positive braid --- p.17 / Chapter 4 --- T-Links and the braid index --- p.28 / Chapter 4.1 --- T-links --- p.30 / Chapter 4.2 --- Symmetries --- p.35 / Chapter 4.3 --- Trip number and the braid index --- p.39 / Chapter 5 --- Modular knots --- p.47 / Chapter 5.1 --- The Modular flow --- p.47 / Chapter 5.2 --- Modular Knots --- p.49 / Bibliography --- p.55

In the 1960s French mathematician George de Rham found a relationship between two invariants of knots. He found that there exist representations of the fundamental group of a knot into a group G of upper right triangular matrices in C with determinant one that is described exactly by the roots of the Alexander polynomial. I extended this result to find that the representations of the fundamental group of a link into G are described by the multivariable Alexander polynomial of the link.

Thesis (doctoral)--Universität Rostock, 1910. / Vita. Includes bibliographical references.

Thesis (Ph.D.)--Tennessee Technological University, 2009. / Title from title page screen (viewed on Feb. 8, 2010). Bibliography: leaf 154.