Searching for Closed Essential Surfaces in Knot Complements with Character Varieties
Talk by Dr. Charles Katerba (Dept of Mathematical Sciences, MSU)
2/20/2019 Wilson Hall 1-144 4:10-5:00pm
Culler-Shalen theory uses a 3-manifold’s (P)SL(2,C) character variety to construct essential surfaces in the manifold. It has been a fundamental tool over the last 35 years in low-dimensional topology. Much of its success is due to a solid understanding of the essential surfaces with boundary that can be constructed with the theory. It turns out, however, that not every surface with boundary is detected. Moreover, one can also construct closed essential surfaces within this framework. In this talk, we will discuss a module-theoretic perspective on Culler-Shalen theory and apply this perspective to show that there are knot complements in S^3 which contain closed essential surfaces, none of which are detected by Culler-Shalen theory. As a corollary, we will construct an infinite family of closed hyperbolic Haken 3-manifolds whose representations into PSL(2, C) have a special number-theoretic property.