Seminars and short courses

Turaev–Viro (TV) invariants are 3-manifold invariants, defined for a given fixed integer $r$ and $2r$-th root of unity. Chen and Yang extended the definition of TV-invariants to pseudo 3-manifolds and introduced a volume conjecture for TV-invariants which states that for the case of $r$-th roots of unity where $r$ is odd and $M$ is hyperbolic, the TV invariants of $M$ grow exponentially and determine the volume of $M$.

The Witten–Reshetikhin–Turaev (WRT) 3-manifold invariants (also known as the Chern–Simons 3-manifold invariants), are defined for a given fixed integer $r$, and a $2r$-th root of unity. The existence of such invariants were predicted by Witten in his work on Chern–Simons gauge theory and topological quantum field theory. They were constructed by Reshetikhin and Turaev by using representation theory and Kirby calculus. Later, Lickorish gave a skein theoretic definition. These invariants were also originally defined for closed orientable 3-manifolds, but were recently extended to link complements. Furthermore, Belletti, Detcherry, Kalfagianni, and Yang provided an explicit formula relating the TV invariant to the WRT invariant of link complements in a closed orientable 3-manifold and used this formula to prove the TV volume conjecture for octahedral link complements in the connected sums of $S^2 \times S^1$ called fundamental shadow links.

In contrast, fully augmented links are links in $S^3$ whose complements have nice geometric properties. For instance, Agol and Thurston showed that fully augmented links can be decomposed into totally geodesic, right-angled ideal polyhedra. In this talk, we will present a geometric description of the relationship between octahedral fully augmented links and fundamental shadow links and we will outline an alternative proof, using the colored Jones polynomial, to prove the TV volume conjecture for octahedral fully augmented links with no half-twists. This is joint work with Emma McQuire and Jessica Purcell.