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

학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2019. 2. 박종일.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.1 Introduction 1 1.1 Deformed Ptolemy assignments . . . . . . . . . . . . . . . . . . . 1 1.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Potential functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Cluster variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Preliminaries 12 2.1 Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Obstruction classes . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Ptolemy varieties 16 3.1 Formulas of Neumann . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Deformed Ptolemy varieties . . . . . . . . . . . . . . . . . . . . . 19 3.2.1 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.2 Pseudo-developing maps . . . . . . . . . . . . . . . . . . . 27 3.3 Flattenings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.1 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 Potential functions 43 4.1 Generalized potential functions . . . . . . . . . . . . . . . . . . . 43 4.1.1 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . 45 4.2 Relation with a Ptolemy assignment . . . . . . . . . . . . . . . . 50 4.2.1 Proof of Theorem 4.2.1 . . . . . . . . . . . . . . . . . . . 54 4.3 Complex volume formula . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.1 Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . 59 5 Cluster variables 70 5.1 The Hikami-Inoue cluster variables . . . . . . . . . . . . . . . . . 70 5.1.1 The octahedral decomposition . . . . . . . . . . . . . . . 70 5.1.2 The Hikami-Inoue cluster variables . . . . . . . . . . . . . 71 5.1.3 The obstruction cocycle . . . . . . . . . . . . . . . . . . . 74 5.1.4 Proof of Theorem 1.3.2 . . . . . . . . . . . . . . . . . . . 75 5.2 The existence of a non-degenerate solution . . . . . . . . . . . . . 79 5.2.1 Proof of Proposition 5.2.1 . . . . . . . . . . . . . . . . . . 81 5.2.2 Explicit computation from a representation . . . . . . . . 83Docto