On the characteristic and deformation varieties of a knot
| dc.creator | Garoufalidis, Stavros | |
| dc.date | 2003-06-15 | |
| dc.date | 2004-09-20 | |
| dc.date.accessioned | 2026-07-07T04:58:58Z | |
| dc.date.available | 2026-07-07T04:58:58Z | |
| dc.description | 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 C^2. We then compare it with the character variety of SL_2(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. | |
| dc.description | Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper12.abs.html | |
| dc.identifier | https://arxiv.org/abs/math/0306230 | |
| dc.identifier | http://arxiv.org/abs/math/0306230 | |
| dc.identifier | Geom. Topol. Monogr. 7 (2004) 291-309 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/67799 | |
| dc.subject | Geometric Topology | |
| dc.subject | Quantum Algebra | |
| dc.subject | 57N10, 57M25 | |
| dc.title | On the characteristic and deformation varieties of a knot | |
| dc.type | text |