Conway spheres as ideal points of the character variety

Detecting Essential Surfaces as Intersections in the Character Variety

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.

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...

The ideal of the trifocal variety

Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor is a trifocal tensor is also given.

Towards the boundary of the character variety

The ideal glass transition of hard spheres.

We use the replica method to study the ideal glass transition of a liquid of identical hard spheres. We obtain estimates of the configurational entropy in the liquid phase, of the Kauzmann packing fraction phi(K), in the range of 0.58-0.62, and of the random close packing density phi(c), in the range of 0.64-0.67, depending on the approximation we use for the equation of state of the liquid. We...