Volume of hyperbolic 3-manifolds

introduction

• volume is an important invariant of hyperbolic 3-manifold
• big open problem Kashaev's volume conjecture
• three simple hyperbolic knots
  • \(4_{1}\) figure 8 knot
  • \(5_{2}\)
  • \(6_{1}\), \(6_{1}\), \(6_{1}\)
• A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3-manifold is bounded below by a linear function of its Heegaard genus

volume of figure eight knot complement

• Figure eight knot

other examples

• \(V(4_{1})=2.029883212819\cdots\)
• \(V(5_{2})=2.82812208\cdots\)
• \(V(6_{1})=3.163963228\cdots\)

Chern-Simons invariant

• Chern-Simons theory

links

• 매듭이론 (knot theory)

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxX3ZsSC04OEUwU0k/edit

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)
• http://en.wikipedia.org/wiki/

expositions

• Steven Finch, Volumes of Hyperbolic 3-Manifolds, September 5, 2004 http://algo.inria.fr/csolve/hyp.pdf

articles

• Purcell, Jessica S., and Alexander Zupan. “Independence of Volume and Genus \(g\) Bridge Numbers.” arXiv:1512.03869 [math], December 11, 2015. http://arxiv.org/abs/1512.03869.
• Le, Thang. “Growth of Homology Torsion in Finite Coverings and Hyperbolic Volume.” arXiv:1412.7758 [math], December 24, 2014. http://arxiv.org/abs/1412.7758.
• Alexander Goncharov, Volumes of Hyperbolic Manifolds and Mixed Tate Motives, 1999
• Gliozzi, F., and R. Tateo. 1995. Thermodynamic Bethe Ansatz and Threefold Triangulations. hep-th/9505102 (May 17). doi:doi:10.1142/S0217751X96001905. http://arxiv.org/abs/hep-th/9505102.
• Adams, C., Hildebrand, M. and Weeks, J., Hyperbolic invariants of knots and links, Trans. Amer.Math. Soc. 1 (1991), 1–56.
• Don Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월
• Borel, A. “Commensurability Classes and Volumes of Hyperbolic 3-Manifolds.” Ann. Sc. Norm. Super. Pisa8, 1–33 (1981)

http://www.numdam.org/numdam-bin/item?ma=211807&id=ASNSP_1981_4_8_1_1_0.