Exact Computation of the n-Loop Invariants of Knots

Abstract

The loop invariants of Dimofte–Garoufalidis is a formal power series with arithmetically interesting coefficients that conjecturally appears in the asymptotics of the Kashaev invariant of a knot to all orders in 1/N. We develop methods implemented in SnapPy that compute the first six coefficients of the formal power series of a knot. We give examples that illustrate our method and its results.

Keywords:

• knots
• Jones polynomia
• Kashaev invariant
• volume conjecture
• volume
• hyperbolic geometry
• ideal triangulations
• shapes
• Neumann–Zagier data
• 1-loop
• n-loop
• formal Gaussian integration
• Feynman diagrams

2000 AMS Subject Classification:

• Primary 57N10
• Secondary 57M25

Funding

S.G. was supported in part by NSF.

Acknowledgments

The authors wish to thank the computer support group of the School of Mathematics and especially Justin Filoseta and Lew Lefton. We also wish to thank the anonymous referee for a careful reading of our article and for suggested improvements in the presentation.