On the characteristic and deformation varieties of a knot

By Stavros Garoufalidis. 2003

231 citations recorded by the source corpus.

The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose nth term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of sl2. 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 in- troduce 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 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. AMS Classification 57N10; 57M25

Subfields: Computational Mechanics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/804e4cbabdabf97559f59555303bee604a6468ae

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Geometry and Topology

Citations in the corpus

DIFFERENCE EQUATION OF THE COLORED JONES POLYNOMIAL FOR TORUS KNOT (2004, background): “Motivated by “volume conjecture” [12, 18] saying that a hyperbolic volume of the knot complement dominates an asymptotic behavior of the colored Jones polynomial JK (N) at q = e2πi/N, it receives much interests toward a geometrical and topological interpretation of the quantum invariants.”
3-Manifolds and 3d Indices (2011, background): “7) is well known in knot theory and Chern-Simons theory as the “quantum A-polynomial” of the figure-eight knot [29, 31, 52].”
Quantum Riemann Surfaces in Chern-Simons Theory (2011, methodology): “Such a recursion relation for JN(K; q) was found quite independently of analyticallycontinued Chern-Simons theory in [5, 6].”
Chern-Simons theory and S-duality (2011, background): “For example, for an (a, b) torus knot, the quantum Â-polynomial and its S-dual are given by28 [11, 85, 86]”
A-polynomial, B-model, and quantization (2011, methodology): “) In this context, the quantum operator Â is usually hard to construct (see [40, 41] for first indirect calculations and [25] for the most recent and systematic ones); therefore, any insight offered by an alternative method is highly desirable.”
Exact results for perturbative Chern-Simons theory with complex gauge group. (2009, methodology): “Note that [17, 18] look at asymptotics of the colored Jones polynomial normalized by its value at the unknot, while the SL(2, C) Chern-Simons partition function should agree with the unnormalized colored Jones polynomial.”
Gauge Theories Labelled by Three-Manifolds (2011, background): “In Chern-Simons theory, (6.25) is known as the generalized / quantum volume conjecture [8] (sometimes also called the AJ-conjecture [56, 57] in the math literature), whereas in N = 2 gauge theory it expresses Ward identities for line operators.”
The colored Jones polynomial and the A-polynomial of Knots☆ (2004, background): “Garoufalidis [11] noticed that by adding to T all the inverses of polynomials in M one gets a principal ideal domain T̃ , and hence from the ideal AK one can define a polynomial invariant.”