The colored Jones polynomial and its stability

Colored Jones polynomial;Planar graph;Q-series;Flag algebra

Vuong, Thao Minh ; Garoufalidis, Stavros Mathematics Etnyre, John Le, Thang T.Q. Margalit, Dan Tran, Anh ; Garoufalidis, Stavros

University:Georgia Institute of Technology

Department:Mathematics

关键词: Colored Jones polynomial;    Planar graph;    Q-series;    Flag algebra;   

Others  :  https://smartech.gatech.edu/bitstream/1853/52212/1/VUONG-DISSERTATION-2014.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

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 conjecturefor the coefficients of the colored Jones polynomial of a knot, coloredby irreducible representations in afixed ray of a simple Lie algebra. The conjecture is verified for all torus knots andall 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 allgraphs with at most 8 edges.In addition, a graph-theory proof of a theorem of Dasbach-Linwhich identifies the coefficient of q^kin those series for k=0,1,2 in terms of polynomials on the number ofvertices, 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 fourstable coefficients are identified with polynomial invariants of a (reduced) Tait graph ofthe link projection. A free polynomial algebraof invariants of graphs whose elements give invariants of alternating links is introducedwhich strictly refines the first four stable coefficients. It is conjectured that all stable coefficients are elements of this algebra, and experimentalevidence 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 allirreducible planar graphs with at most 6 vertices.