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1	The volume conjecture, the aj conjectures and skein modules Tran, Anh Tuan 21 June 2012 (has links) This dissertation studies quantum invariants of knots and links, particularly the colored Jones polynomials, and their relationships with classical invariants like the hyperbolic volume and the A-polynomial. We consider the volume conjecture that relates the Kashaev invariant, a specialization of the colored Jones polynomial at a specific root of unity, and the hyperbolic volume of a link; and the AJ conjecture that relates the colored Jones polynomial and the A-polynomial of a knot. We establish the AJ conjecture for some big classes of two-bridge knots and pretzel knots, and confirm the volume conjecture for some cables of knots. Volume conjecture AJ conjecture Skein module Colored Jones polynomial A-polynomial Knot theory Symmetry (Mathematics) Invariants Invariant manifolds Hyperbolic spaces