Yokota theory, the invariant trace fields of hyperbolic knots and the Borel regulator map

TL;DR

This paper explores the relationship between hyperbolic knot invariants, the solutions of hyperbolicity equations, and the Borel regulator map, revealing new connections and confirming them for twist knots.

Contribution

It establishes a link between the number of essential solutions of hyperbolicity equations and the invariant trace field extension degree, and relates the potential function to the Borel regulator map.

Findings

Number of essential solutions ≥ extension degree of invariant trace field

Potential function encodes complex volumes and Borel regulator values

Maximum imaginary part of complex volume equals hyperbolic volume

Abstract

For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the invariant trace field of the link. On the other hand, Yokota suggested a potential function of a hyperbolic knot, which gives the hyperbolicity equations and the complex volume of the knot. Applying the fact above to his theory, we explain that the potential function also gives all the values of the Borel regulator map and the complex volumes of the parabolic representations. Furthermore, we explain the maximum value of the imaginary parts of the complex volumes is the volume of the complete hyperbolic structure of the knot complement. Especially, if the number of the essential solutions of the hyperbolicity equations and the extension degree of the invariant trace field are the same, then the evaluation of all essential complex solutions of the hyperbolicity equations to the imaginary part of the potential function is the same with the Borel regulator map. We show these actually happens in the case of the twist knots.