  We calculate the knot invariant coming from the Teichm\"{u}ller TQFT [AK1].
Specifically we calculate the knot invariant for the complement of the knot
$6_1$ both in the original [AK1] and the new formulation of the Teichm\"{u}ller
TQFT [AK2] for the one-vertex H-triangulation of $(S^3,6_1)$. We show that the
two formulations give equivalent answers. Furthermore we apply a formal
stationary phase analysis and arrive at the Andersen- Kashaev volume conjecture
as stated in [AK1, Conj. 1]. Furthermore we calculate the first examples of
knot complements in the new formulation showing that the new formulation is
equivalent to the original one in all the special cases considered. Finally, we
provide an explicit isomorphism between the Teichm\"{u}ller TQFT representation
of the mapping class group of a once punctured torus and a representation of
this mapping class group on the space of Schwartz class functions on the real
line.
