Twist knot intersections in the character variety of the Whitehead link

Charles Katerba (Montana State)

Abstract

There is a large body of mathematical knowledge dedicated to determining ways to distinguish mathematical objects. This talk will address a sort of converse question: when two objects are known to be distinct, how similar are they? The question is ill-posed in general, but can be made more concrete in certain cases. For example, if two three manifolds \(M\) and \(N\) arise from Dehn filling one component of a \(2\)-component link in \(S^3\), both of their fundamental groups can be realized as quotients of the same group. We can then use character varieties to study similarities between the fundamental groups of \(M\) and \(N\). We implement this idea for a very concrete class of examples: the twist knots. These knots arise from Dehn filling one component of the Whitehead link and their character varieties can be realized as curves in the character variety of the Whitehead link. We study the pairwise intersection of these curves, classify an arithmetic property of points in the intersection, and interpret this arithmetic property topologically.