The volume and Chern-Simons invariant of a Dehn-filled manifold

Abstract

Based on the work of Neumann, Zickert gave a simplicial formula for computing the volume and Chern-Simons invariant of a boundary-parabolic $\psl$-representation of a compact 3-manifold with non-empty boundary. Main aim of this thesis is to introduce a notion of deformed Ptolemy assignments (or varieties) and generalize the formula of Zickert to a representation of a Dehn-filled manifold. We also generalize the potential function of Cho and Murakami by applying our formula to an octahedral decomposition of a link complement in the 3-sphere. Also, motivated from the work of Hikami and Inoue, we clarify the relation between Ptolemy assignments and cluster variables when a link is given in a braid position. The last work is a joint work with Jinseok Cho and Christian Zickert.