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Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knot. (English) Zbl 1092.57005
The Volume Conjecture for hyperbolic knots consists of two parts: (a) it states that the limit of a sequence of complex numbers (involving the \(n\)-th colored Jones polynomial evaluated at the primitive complex \(n\)-th root of unity) exists and (b) it identifies the limit essentially with the hyperbolic volume of the knot. The Volume Conjecture is known only for the simplest hyperbolic knot \(4_1\). The authors develop an efficient formula for the colored Jones function of the \(k4_3\) knot, which is the simplest hyperbolic non \(2\)-bridge knot. Then using that formula they present numerical evidence (via a Fortran program) for the Volume Conjecture for the \(k4_3\) knot. For this knot they also provide among other computations a presentation of its fundamental group and peripheral system, the Alexander and \(A\)-polynomial, and the rank of the Heegaard-Floer Homology.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
Software:
qZeil; Snap; SnapPea
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