Thesis advisor: John A. Baldwin / We study the relationship between knots in contact three-manifolds and complex curves in Stein surfaces. To do so, we extend the notion of quasipositivity from classical braids to links that are braided with respect to an open book decomposition of an arbitrary closed, oriented three-manifold. Our main results characterize the transverse links in Stein-fillable contact three-manifolds that bound smooth holomorphic curves in Stein fillings. This characterization is made possible by new techniques in the theory of characteristic and open book foliations on surfaces in three-manifolds. We also explore the Seifert genera of cross-sections of complex plane curves, minimal braid representatives of quasipositive links, and the relationship between Legendrian ribbons in contact three-manifolds and strongly quasipositive braids with respect to compatible open books. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis (Ph.D.)--Michigan State University. Mathematics, 2008. / Title from PDF t.p. (viewed on July 24, 2009) Includes bibliographical references (p. 40-41). Also issued in print.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 67-69). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 129-135).

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Thesis advisor: Tao Li / This dissertation explores a relationship between fibered knots and Heegaard splittings in closed, connected, orientable three-manifolds. We show that a fibered knot, which has a sufficiently complicated monodromy, induces a minimal genus Heegaard splitting that is unique up to isotopy. Moreover, we show that fibered knots in the three-sphere has complexity at most 3. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

This thesis covers improvements on the upperbounds for ropelength of a specific class of algebraic knots.

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We will discuss what is known about the construction of contact structures via branched covers, emphasizing the search for universal transverse knots. Recall that a topological knot is called universal if all 3-manifold can be obtained as a cover of the 3-sphere branched over that knot. Analogously one can ask if there is a transverse knot in the standard contact structure on S³ from which all contact 3-manifold can be obtained as a branched cover over this transverse knot. It is not known if such a transverse knot exists.