On the integral of $\log x\frac{dy}{y}-\log y\frac{dx}{x}$ over the A-polynomial curves

Details On the integral of $\log x\frac{dy}{y}-\log y\frac{dx}{x}$ over the A-polynomial curves On the integral of $\log x\frac{dy}{y}-\log y\frac{dx}{x}$ over the A-polynomial curves

2008-11-17

arxiv.org/abs/0811.2725v1

Mathematics

Geometric Topology

To appear in the Proceedings of the International Conference on Quantum Topology

Scientific paper

In this note, we study the integral of the 1-form $\log x\frac{dy}{y}-\log y\frac{dx}{x}$ over certain plane curves defined by A-polynomials of knots. It is quite surprising that a Chern-Simons type invariant of 3-manifolds, which can be geometrically computed, may be used to get the exact values of those integrals. The arithmetic nature of these integrals is still unknown at the moment and deserved further investigation.

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