The Geometry and Topology of the Jones Polynomial

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In 1985 Jones assigned to a knot a polynomial with integer coefficients,and in 1989 Witten came up with a path-integral interpretation of thispolynomial. Upon taking parallels, one can consider a sequence of polynomials. The volume conjecture relates an asymptotic behavior of thissequence (evaluated at complex roots of unity) with the Riemannian (mostlyhyperbolic) geometry of the knot complement. The PIs propose to study thisand related conjectures that relate quantum topology and geometrization.A knot in 3-space is a flexible rope that is allowed to deform withoutcrossing itself. The width and length of the rope are irrevelant, and itsbeginning and end coincide. For example, a rubber band is an example of a knot. In the middle of 19th century, Kelvin postulated that chemical elements (such as gold, aluminum) are made out of knots; thus differentknot types explain the variation of observed chemical elements. More than a century later, string-theory offers a similar view of the world, whereelementary particles are vibrating in the fabric of cosmos, along knottedcircles. Whether string-theory explains the world is a key question. Meanwhile, it is known that string-theory has deeply influenced mathematics.