Instrumental Commensurability

Commensurability Classes of Twist Knots

In this paper we prove that if MK is the complement of a non-fibered twist knot K in S, then MK is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In ad...

Traces, lengths, axes and commensurability

This paper is based on three lectures given by the author at the workshop, “Hyperbolic geometry and arithmetic: a crossview” held at The Université Paul Sabatier, Toulouse in November 2012. The goal of the lectures was to describe recent work on the extent to which various geometric and analytical properties of hyperbolic 3-manifolds determine the commensurability class of such manifolds. This ...

Commensurability riteria for Kleinian groups

Mixing methodologies and paradigmatic commensurability

My first critical awareness of the tensions between different methodologies arose 5 years ago when preparing the materials for two undergraduate courses: ‘Operations Management’ and ‘Strategy’. The former was concerned with explanation in which there is the assumption of causality; the operations environment could be designed in an optimal manner with predictable behaviour—the domain of those f...

Commensurability and the character variety

Recall that hyperbolic 3-manifolds M and N are said to be commensurable if they have a common finite sheeted covering. This is equivalent to the fundamental groups having subgroups of finite index which are conjugate in PSL(2,C). In general it is very difficult to determine if two manifolds are commensurable or not, once the most obvious invariants of commensurability (for example, the invarian...