Closed surfaces and character varieties

Abstract

The powerful character variety techniques of Culler and Shalen can be used to find essential surfaces in knot manifolds. We show that module structures on the coordinate ring of the character variety can be used to identify detected boundary slopes as well as when closed surfaces are detected. This approach also yields new number theoretic invariants for the character varieties of knot manifolds.Comment: 28 pages, 1 figur

Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3–manifolds

“…Finally, Chesebro showed that if X SL pN q ANI-detects a closed essential surface, then it detects one by ideal points [6,Proposition 5.2]. This contradicts our hypothesis, so N pp i {q i q has no ANI-representations into PSL 2 pCq.…”

“…Chesebro proved that if an irreducible curve in X SL pN q ANI-detects a closed essential surface, then it detects a closed essential surface with ideal points [6,Proposition 5.2]. Goodman, Heard, and Hodgson compiled a partial list of all hyperbolic knots and links with up to 12 crossings whose discrete and faithful representations are ANI.…”

“…Chesebro [6] noticed a connection between the detection of essential surfaces by ideal points and an infinite collection of modules associated to the coordinate ring CrXs of X which we now describe.…”

“…The proof of Theorem 1 uses a module-theoretic approach to Culler-Shalen theory developed by Chesebro [6]. As will be explained further in §4, this approach transforms the problem into a computation in commutative algebra.…”

“…Thus, we are able to answer this question using algorithmic techniques implemented in the software package Macaulay2 [15]. It also follows from [6,Proposition 5.2] that no closed essential surface in the complement of 10 152 , 10 153 or 10 154 is ANI-detected.…”