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 S¹ x S², 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 3-sphere. The complements of these links are now known to be homeomorphic to complements of fundamental shadow links, using topological techniques. 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.

This research was partially supported by the Australian Government through the Australian Research Council (ARC) Discovery Projects funding scheme, projects DP210103136 and DP240102350, and by a Research Training Program (RTP) Scholarship doi.org/10.82133/C42F-K220.