Octahedral developing of knot complement I: pseudo-hyperbolic structure

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.

55 pages, 31 figures

References in corpus (2)

• Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds
• Octahedral developing of knot complement II: Ptolemy coordinates and applications

Cited by in corpus (9)

• Quantum holonomic link invariants derived from stated skein algebras
• Octahedral developing of knot complement II: Ptolemy coordinates and applications
• The volume and Chern-Simons invariant of a Dehn-filled manifold
• On the potential functions for a link diagram
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket
• ${\mathrm{SL}}_2(\mathbb{C})$-holonomy invariants of links
• Octahedral coordinates from the Wirtinger presentation
• On the volume and the Chern-Simons invariant for the hyperbolic alternating knot orbifolds
• Symplectic quandles and parabolic representations of 2-bridge Knots and Links