Cyclic surgery on satellite knots in S³ is classified and a necessary condition is given for a knot in S³ to admit a nontrivial cyclic surgery with slope m/l, \m\ > 1. A complete classification of cyclic group actions on the Poincaré sphere with 1-dimensional fixed point sets is obtained. It is proved that the following knots have property I, i.e. the fundamental group of the manifold obtained by Dehn surgery on such a knot cannot be the binary icosahedral group I₁₂₀, the fundamental group of the Poincaré homology 3-sphere: nontrefoil torus knots, satellite knots, nontrefoil generalized double knots, periodic knots with some possible specific exceptions, amphicheiral strongly invertible knots, certain families of pretzel knots. Further the Poincaré sphere cannot be obtained by Dehn surgery on slice knots and a certain family of knots formed by band-connect sums. It is proved that if a nonsufficiently large hyperbolic knot in S³ admits two nontrivial cyclic Dehn surgeries then there is at least one nonintegral boundary slope for the knot. There are examples of such knots. Thus nonintegral boundary slopes exist. / Science, Faculty of / Mathematics, Department of / Graduate

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A knot invariant called the Jones polynomial will be defined in two ways, as the Kauffman Bracket polynomial and the Tutte polynomial. Three properties of the Jones polynomial are discussed. We also see how mutant knots share the same Jones polynomial.

This thesis reviews the history of knot theory with an emphasis on the diagrammatic approach to studying knots. Also covered are the basic concepts and notions of graph theory and how these two fields are related with an example of a knot diagram and how to associate it to a graph.

In this thesis, we show that every periodic virtual knot can be realized as the closure of a periodic virtual braid. If K is a q-periodic virtual knot with quotient K_*, then the knot group G_{K_*} is a quotient of G_K and we derive an explicit q-symmetric Wirtinger presentation for G_K, whose quotient is a Wirtinger presentation for G_{K_*}. When K is an almost classical knot and q=p^r, a prime power, we show that K_* is also almost classical, and we establish a Murasugi-like congruence relating their Alexander polynomials modulo p. This result is applied to the problem of determining the possible periods of a virtual knot $K$. For example, if K is an almost classical knot with nontrivial Alexander polynomial, our result shows that K can be p-periodic for only finitely many primes p. Using parity and Manturov projection, we are able to apply the result and derive conditions that a general q-periodic virtual knot must satisfy. The thesis includes a table of almost classical knots up to 6 crossings, their Alexander polynomials, and all known and excluded periods. / Thesis / Doctor of Philosophy (PhD)

Doctor of Philosophy / Department of Mathematics / David Yetter / In this paper we give a new generalization of the Khovanov homology. The construction begins with a Frobenius-algebra-like object in a category of graded vector-spaces with an anyonic braiding, with most of the relations weaken to hold only up to phase. The construction of Khovanov can be adapted to give a new link homology theory from such data. Both Khovanov's original theory and the odd Khovanov homology of Oszvath, Rassmusen and Szabo arise from special cases of the construction in which the braiding is a symmetry.

Thesis (M.S.)--University of Nevada, Reno, 2006. / "May, 2006." Includes bibliographical references (leaf 40). Online version available on the World Wide Web.

Thesis (M.S.)--Duquesne University, 2005. / Title from document title page. Abstract included in electronic submission form. Includes bibliographical references (p. 31) and index.

Honors Project--Smith College, Northampton, Mass., 2010. / Includes bibliographical references (p. 64-65).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1983. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 52-55).