Mathematics – Geometric Topology
Scientific paper
2010-03-25
Contemporary Mathematics 541 (2011), p.41-67
Mathematics
Geometric Topology
32 pages, 6 figures; acknowledgements updated
Scientific paper
The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, nontrivial manner. The goal of the present lectures is to review the original statement of the volume conjecture and its recent extensions and generalizations, and to show how, in the most general context, the conjecture can be understood in terms of topological quantum field theory. In particular, we consider: a) generalization of the volume conjecture to families of incomplete hyperbolic metrics; b) generalization that involves not only the leading (volume) term, but the entire asymptotic expansion in 1/N; c) generalization to quantum group invariants for groups of higher rank; and d) generalization to arbitrary links in arbitrary three-manifolds.
Dimofte Tudor
Gukov Sergei
No associations
LandOfFree
Quantum Field Theory and the Volume Conjecture does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Quantum Field Theory and the Volume Conjecture, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quantum Field Theory and the Volume Conjecture will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-555908