2003/06/30 by Stavros Garoufalidis · 2 citations
Mathematics · #math.GT #math.QA #msc:57N10 #msc:57M25
published as Geom. Topol. Monogr. 7 (2004) 291-309 · Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper12.abs.html
arxiv created 2004/09/20 · arxiv updated 2009/11/30
The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose n-th term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of SL(2). It was recently shown by TTQ Le and the author that the colored Jones function of a knot is q-holonomic, ie, that it satisfies a nontrivial linear recursion relation with appropriate coefficients. Using holonomicity, we introduce a geometric invariant of a knot: the characteristic variety, an affine 1-dimensional variety in C2. We then compare it with the character variety of SL2(C) representations, viewed from the boundary. The comparison is stated as a conjecture which we verify (by a direct computation) in the case of the trefoil and figure eight knots. We also propose a geometric relation between the peripheral subgroup of the knot group, and basic operators that act on the colored Jones function. We also define a noncommutative version (the so-called noncommutative A-polynomial) of the characteristic variety of a knot. Holonomicity works well for higher rank groups and goes beyond hyperbolic geometry, as we explain in the last chapter.