## The colored Jones polynomial and its stability

##### Abstract

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.