Farey recursion and the character varieties for 2-bridge knots

Abstract

We describe the (P)SL 2 C character varieties of all 2-bridge knots and the diagonal character varieties for all 2-bridge links in terms of a set of polynomials defined using Farey recursion.

Farey Recursive Functions

“…Take x, y, and z to be the traces of the matrices ρ(a), ρ(b), and ρ(c) where a, b, and c are primitive elements of G with respective slopes 0/1, 1/0, and 1/1. Then, as shown in [9], the value of the specialization of U 1 (p/q) is the trace of ρ(g) where g is a primitive element of G with slope p/q. Example 5.4 (Markov numbers).…”

“…Example 5.2 (Generic FRFs). As defined in [9], the generic The generic FRF with determinant d is the FRF U d : Q → Z[x, y, z] as defined above with determinant d (i.e., as usual d 1 = d and d 2 = U). As above, every FRF with image in a ring R and determinant d is a specialization of U d .…”

“…Example 5.5 (Character varieties and Riley polynomials for 2-bridge links). From [9], let T : Q → Z[x, z] be the FRF with determinant one obtained from U 1 by setting y = 0. The main theorems of [9] show that the affine set…”

“…From [9], let T : Q → Z[x, z] be the FRF with determinant one obtained from U 1 by setting y = 0. The main theorems of [9] show that the affine set…”

“…Certain Farey recursive functions arise naturally in the geometry and topology of lowdimensional manifolds and in number theory, see [6], [8], [9], [16] and [22]. This paper is motivated by these works, and its purpose is to introduce the general definition of a Farey recursive function as an interesting object in its own right, provide some interesting examples, and to establish some basic properties.…”