Augmented links, shadow links, and the TV volume conjecture: a geometric perspective

TL;DR

This paper provides a geometric proof that certain link complements are isometric and verifies the Turaev-Viro volume conjecture for these links using skein theory and colored Jones polynomials.

Contribution

It offers a new geometric proof of the equivalence of octahedral fully augmented links and fundamental shadow links, and verifies the volume conjecture for these links.

Findings

Complement of octahedral fully augmented links are isometric to fundamental shadow links.

Derived formulas for colored Jones polynomials of these links.

Confirmed the Turaev-Viro volume conjecture for these links in specific cases.

Abstract

For hyperbolic 3-manifolds, the growth rate of their Turaev-Viro invariants, evaluated at a certain root of unity, is conjectured to give the hyperbolic volume of the manifold. This has been verified for a handful of examples and several infinite families of link complements, including fundamental shadow links. Fundamental shadow links lie in connected sums of copies of $S^1\times S^2$, and their complements are built of regular ideal octahedra. Another well-known family of links with complements built of regular ideal octahedra are the octahedral fully augmented links in the 3-sphere. The complements of these links are now known to be homeomorphic to complements of fundamental shadow links, using topological techniques. In this paper, we give a new, geometric proof that complements of octahedral fully augmented links are isometric to complements of fundamental shadow links. We then use skein theoretic techniques to determine formulae for coloured Jones polynomials of these links. In the case of no half-twists, this gives a new, more geometric verification of the Turaev-Viro volume conjecture for these links.