Proof of the Strong AJ Conjecture for the Figure 8 Knot




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The AJ conjecture, formulated by Garoufalidis [7], relates the A-polynomial of a knot and the colored Jones polynomial of a knot. The strong AJ conjecture first proposed in [6] and then modified by Sikora [14], relates the orthogonal ideal to the classical peripheral ideal. The orthogonal ideal is an ideal of the skein module of the torus and the classical peripheral ideal is an ideal of the coordinate ring of the SL(2, C) character variety. This conjecture could be seen as the topological and algebraic structure that underlies the AJ conjecture. The strong AJ conjecture has been confirmed for all torus knots and cables over torus knots. As such, the conjecture has only been confirmed for cases of non-hyperbolic knots. It should be noted that most knots fall into the class of being hyperbolic. In this thesis we confirm the strong AJ conjecture for the figure 8 knot which is the simplest hyperbolic knot.



Quantum theory, Topology, Modules (Algebra), Knot polynomials, Torus (Geometry), Geometry, Hyperbolic


©2019 Hoang-An A. Nguyen