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This dissertation studies the colored Jones polynomial of knots and links, colored by representations of simple Lie algebras, and the stability of its coefficients. Chapter 1 provides an explicit formula for the second plethysm of an arbitrary representation of sl3. This allows for an explicit formula for the colored Jones polynomial of the trefoil, and more generally, for T(2,n) torus knots. This formula for the sl3 colored Jones polynomial of T(2,n)$ torus knots makes it possible to verify the Degree Conjecture for those knots, to efficiently compute the sl3 Witten-Reshetikhin-Turaev invariants of the Poincare sphere, and to guess a Groebner basis for the recursion ideal of the sl3 colored Jones polynomial of the trefoil. Chapter 2 gives a formulation of a stability conjecture for the coefficients of the colored Jones polynomial of a knot, colored by irreducible representations in a fixed ray of a simple Lie algebra. The conjecture is verified for all torus knots and all simple Lie algebras of rank 2. Chapter 3 supplies an efficient method to compute those q-series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges. In addition, a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of q^k in those series for k=0,1,2 in terms of polynomials on the number of vertices, edges and triangles of the graph is given. Chapter 4 provides a study of the structure of the stable coefficients of the Jones polynomial of an alternating link.The first four stable coefficients are identified with polynomial invariants of a (reduced) Tait graph of the link projection. A free polynomial algebra of invariants of graphs whose elements give invariants of alternating links is introduced which strictly refines the first four stable coefficients. It is conjectured that all stable coefficients are elements of this algebra, and experimental evidence for the fifth and sixth stable coefficient is given. The results are illustrated in tables of all alternating links with at most 10 crossings and all irreducible planar graphs with at most 6 vertices.

Er ei fod yn lletchwith a hirwyntog ar brydiau defnyddir yr enw llawn John Gwilym Jones trwy gydol y drafodaeth. Buasai defnyddio John neu John Gwilym yn awgymru agwedd bersonol anghymwys; buasai Mr neu Dr Jones neu J Gwilym Jones yn orffurfiol; a hyfdra dibarch fuasai defnyddio JGJ (ond feli defnyddir yn y nodiadau). Hepgorir teitlau (Mr, Dr, Yr Athro, Syr) yn gyfan gwbi. Er mwyn gwahaniaethu rhwng defnydd John Gwilym Jones a llenorion eraill o goll geiriau ( ... ) yn eu testunau alm toriadau fy hun defnyddiais goll geiriau rhwng bachau petryal i ddynodi fy nhoriadau fy hun fel rheol. Cedwais ddyfyniadau o'r Saesneg yn yr iaith wreiddiol a chyfieithiadau o ieithoedd eraill ilr Gymraeg neulr Saesneg yn iaith y cyfieithiad.

Thesis (S.T.L.)--Catholic University of America, 2006. / Includes bibliographical references (leaves 87-88).

Thesis (Ph.D.)--Indiana University, Dept. of Curriculum and Instruction, 2006. / Adviser: Enid Zimmerman.

Thesis (Ph. D.)--University of New South Wales, 2001. / Also available online.

Chemnitz, Techn. Univ., Masterarb., 2008.

Thesis (M.A.)--Villanova University, 2008. / English Dept. Includes bibliographical references.

Thesis (M.A.)--Ohio University, June, 2009. / Title from PDF t.p. Release of full electronic text on OhioLINK has been delayed until June 1, 2014. Includes bibliographical references (leaves 64-69)