Detecting essential surfaces as intersections in the character variety

Abstract

We describe a family of hyperbolic knots whose character variety contain exactly two distinct components of characters of irreducible representations. The intersection points between the components carry rich topological information. In particular, these points are non-integral and detect a Seifert surface.arXiv:1609.04780v2 [math.GT]

Arithmetic of the canonical component of the Knot $7_4$

“…Also, note that this presentation for the fundamental group comes from the two-bridge normal form for 7 4 , namely (15,11). These facts and the following are in [5], section 5.…”

“…Remark 6.15 (Affine Intersection Points). Chu showed in [5] Section 5.1 that there are four affine points on the canonical component, C, for 7 4 which intersect the other component of the character variety containing the character of an irreducible representation. They lie in a number field L of degree 4.…”

“…The focus of this paper is arithmetic and algebraic properties of a particular component C (the canonical component, see Section 2.1) of X(Γ) when Γ is the fundamental group of a particular hyperbolic knot complement (7 4 of the tables of [16] and (15,11) in two-bridge notation.). This has already been studied for different reasons ( [5]). There are two themes to this: the first is that, following [4], we are particularly interested in a canonically defined quaternion algebra, A k (C), which is defined over the function field of C, k(C), and specializes at Dehn surgery points of C to quaternion algebras defined over number fields.…”

“…We construct such a sequence by repeatedly applying Lemmas 5.11 and 5.12. Set n 1 = 5 5 . We in fact do have 2 n1+1 Im (β n1 ) < 0.…”

Farey recursion and the character varieties for 2-bridge knots

“…However, being Dehn fillings on a hand full of links, it is quick to check that the volumes for these examples are bounded by 7.4, while the volumes of 2-bridge links can be arbitrarily large. Nonetheless, including Riley's original results, these formulas have enjoyed a variety of applications including [5], [6], [11], [17], [18], [19], and [26].…”