On Intersection of Character Varieties Associated to Dehn Fillings of the Whitehead Link and Its Generalization

Abstract

The canonical component of the character variety of a hyperbolic 2-bridge link complement M is an algebraic surface, from the work of Thurston. If we fix a boundary component T of M and perform a Dehn-filling on T, we get a 1-cusped manifold whose character variety is commonly a generalized curve that is naturally embedded in the character variety of M. If we perform distinct Dehn-fillings on M, we get various generalized curves on the canonical surface. In this paper, we analyze how these curves intersect with each other in the surface and study the topological implications for these intersection points. We also classify the intersection points that correspond to reducible characters, using the twisted Alexander polynomials associated to the Dehn-fillings. Throughout the paper, we use the Whitehead link as a motivating example to illustrate our core methodology, while some generalizations to other 2-bridge links will also be discussed.