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

TL;DR

This paper investigates integrals of a specific differential form over A-polynomial curves related to knots, revealing a surprising connection to Chern-Simons invariants and highlighting the need for further exploration of their arithmetic properties.

Contribution

It uncovers a novel link between integrals over knot-related algebraic curves and 3-manifold invariants, suggesting new avenues for understanding their arithmetic nature.

Findings

Integral values relate to Chern-Simons invariants

Surprising connection between geometry and algebraic integrals

Arithmetic properties of integrals remain unknown

Abstract

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.