Plane curves associated to character varieties of 3-manifolds

Abstract

Consider a compact 3-manifold M with boundary consisting of a single torus. The papers [CS1], [CS2] and [CGLS] discuss the variety of characters of SL 2 (C) representations of π 1 (M ), and some of the ways in which the topological structure of M is reflected in the algebraic geometry of the character variety. We will describe in this paper a certain affine algebraic curve D M which is naturally associated to the character variety of M . In the case that M is the complement of a knot K in a homology 3-sphere we may take the basis B to consist of the longitude and meridian of K. With the usual orientation conventions this basis is well-defined modulo the involution which inverts both the longitude and meridian. It will follow from the construction of D M that if the basis B is obtained from B by inverting both generators then the regular maps p B and p B have the same image. Thus A K := A M,B = A M,B is an invariant of the knot.The polynomial A M,B displays, in a striking way, information about the incompressible surfaces in M . This involves the Newton polygon of A M,B , which is the convex hull of the integer lattice points in the plane whose coordinates arise as degrees of monomials in A M,B . Using the main result of [CS1] we show (Theorem 3.4) that "boundary slopes are boundary slopes," that is that the slope of each side of the Newton polygon of A M,B equals the boundary slope of an incompressible surface in M which is associated to an

Exact results for perturbative Chern–Simons theory with complex gauge group

“…In fact, Y has holonomy group Sp(n)×Sp(n), where n is the quaternionic dimension of M flat (G C , Σ). 8 Notice, while in the rest of the paper we consider only the "holomorphic" sector of the theory (which is sufficient in the perturbative approach), here we write the complete symplectic form on M flat (G C , Σ) that follows from the classical Chern-Simons action (1.1), including the contributions of both fields A andĀ.…”

“…(A generalization to arbitrary values of r is straightforward.) In this case, A(l, m) is the so-called A-polynomial of M , originally introduced in [8], and the system (2.31) consists of a single equation…”

Asymptotics of the colored Jones function of a knot

“…Then for α small enough, there is a unique (up to conjugation) abelian SL 2 (C) representation ρ α that satisfies (7). Abelian representations have 0 volume (see eg, [CCG + 94]). On the other hand, for small enough α, we have ∆ K (e α ) ∼ ∆ K (1) = 1.…”

TheC–polynomial of a knot

“…In [6], it was conjectured that this polynomial (which has to do with representations of the quantum group U q .sl 2 /) specializes at q D 1 to the better known A-polynomial of a knot, which has to do with genuine SL 2 ‫/ރ.‬ representations of the knot complement, Cooper-Culler-Gillet-Long-Shalen [4].…”