Octahedral developing of knot complement I: pseudo-hyperbolic structure

TL;DR

This paper develops a method to compute complex volumes and geometric invariants of knot complements using octahedral decompositions and pseudo-hyperbolic structures, providing explicit formulas and examples.

Contribution

It introduces a new approach to analyze knot complements via segment and region variables, enabling explicit computation of complex volumes and holonomy representations.

Findings

Explicit solutions for $T(2,N)$ and $J(N,M)$ knots.

Demonstration of asymptotic behavior of complex volumes.

Formulas for Wirtinger generators and cusp shapes.

Abstract

It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and region variables which are motivated by the volume conjecture so that we can compute complex volumes of all the boundary parabolic representations explicitly. We investigate the octahedral developing and holonomy representation carefully, and obtain a concrete formula of Wirtinger generators for the representation and also cusp shape. We demonstrate explicit solutions for $T(2,N)$ torus knots, $J(N,M)$ knots and also for other interesting knots as examples. Using these solutions we can observe the asymptotic behavior of complex volumes and cusp shapes of these knots. We note that this construction works for any knot or link, and reflects systematically both geometric properties of the knot complement and combinatorial aspect of the knot diagram.