Hyperbolic structures on link complements, octahedral decompositions, and quantum $\mathfrak{sl}_2$

TL;DR

The work builds a bridge between hyperbolic geometry of link complements and the algebraic framework of quantum groups at roots of unity. Using the octahedral decomposition associated to a link diagram, it introduces χ-colorings as coordinates on the $ ext{SL}_2(\mathbb{C})^*$ representation variety and shows how these colors arise naturally from a root-of-unity braiding in $\mathcal{U}_{\xi}(\mathfrak{sl}_2)$. The paper develops two equivalent, reduced coordinate systems—the $b$-variety (segment equations) and the $a$-variety (region equations)—and demonstrates how to recover the gluing equations of the octahedral decomposition from these data, including explicit treatments of twist regions and torus knots. By embedding the central characters into Weyl-algebra-like coordinates, it clarifies how quantum holonomy invariants at roots of unity encode geometric information about hyperbolic structures, thereby connecting BGPR-type invariants with classical hyperbolic geometry. The results illuminate how nondegenerate χ-colorings yield irreducible hyperbolic structures and provide practical computational tools for exploring representations and A-polynomials via octahedral decompositions.

Abstract

Hyperbolic structures on link complements (equivalently, representations of the fundamental group into $\operatorname{SL}_2(\mathbb{C})$) can be described algebraically by using the octahedral decomposition determined by a link diagram. The decomposition (like any ideal triangulation) gives a set of gluing equations in shape parameters whose solutions are hyperbolic structures. We show that these equations can be obtained from Kashaev-Reshetikhin's braiding on the Kac-de Concini quantum group $\mathcal{U}_ξ(\mathfrak{sl}_2)$ at a root of unity $ξ$. This braiding gives coordinates on the $\operatorname{SL}_2(\mathbb{C})$ representation variety of a link and our work shows how to interpret these geometrically.

Figures (12)

• Figure 1: Positive (left) and negative (right) crossings.
• Figure 2: The path $w_3$ in $\pi_1(M_L)$ and the path $x_1^+ x_2^+ x_3^+ \left(x_3^- \right)^{-1} \left(x_2^+\right)^{-1} \left(x_1^+\right)^{-1}$ in $\Pi_1(D)$ are equivalent.
• Figure 3: The four-term and five-term decompositions of an octahedron.
• Figure 4: Shapes of edges at a positive crossing. There are four horizontal edges at the four corners and four vertical edges below and above the four segments.
• Figure 5: Two examples of the region gluing equations. The product of all the parameters is $1$ regardless of the orientation of the boundary segments.

Theorems & Definitions (88)

• definition 1
• definition 2
• definition 3
• definition 4
• example 5
• definition 6
• proposition 7
• definition 8
• remark 9
• theorem 10