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Columbia Dissertations and Theses
> Doctoral Dissertations
A-polynomial and Bloch invariants of hyperbolic 3-manifolds
| Author(s): | Champanerkar, Abhijit |
| Title: | A-polynomial and Bloch invariants of hyperbolic 3-manifolds |
| Physical Description: | v, 108 leaves, bound. |
| Issue Date: | 2003 |
| Description: | Department: Mathematics.
Thesis (Ph. D.)--Columbia University, 2003. |
| Bookmark as: | http://hdl.handle.net/10022/AC:P:5330 |
| Full Text (ProQuest): | /ac/proxit.jsp?url=http://gateway.proquest.com/ope... |
| Abstract: | Let N be a complete, orientable, finite-volume, one-cusped hyperbolic 3-manifold with an ideal triangulation. Using combinatorics of the ideal triangulation of N we construct a plane curve in CxC which contains the squares of eigenvalues of PSL(2, C ) representations of the meridian and longitude. We show that the defining polynomial of this curve is related to the PSL(2, C ) A-polynomial and has properties similar to the classical A-polynomial. We further show that a factor of this polynomial, A0(l, m), associated to the discrete, faithful representation of pi1(N) in PSL(2, C ) is independent of the ideal triangulation. The Bloch invariant beta( N) of N is related to the volume and Chern-Simons invariant of N. The variation of Bloch invariant is defined to be the change of beta(N) under Dehn surgery on N. We relate A0(l, m) to the variation of the Bloch invariant of N. We show that A0(l, m) determines the variation of Bloch invariant in the case when A0( l, m) is a defining equation of a rational curve. We also show that in this case the Bloch invariant reads the symmetry of A 0(l, m). In the next part we classify all hyperbolic knots whose complement can be ideally triangulated with 7 tetrahedra. We obtain 129 knot complements out of 3552 orientable, cusped hyperbolic 3-manifolds with 7 tetrahedra and explicitly describe the corresponding knots in S3. |
| Collection(s): | Doctoral Dissertations
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