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We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. In the case that the 3manifold is a thickened surface, this Kauffman bracket function leads to a Jonestype polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2sided linear bounds on the volume of hyperbolic alternating links in the thickened surface. As a corollary of the proof of this result, we deduce that the twist number of a reduced, twist reduced, checkerboard alternating link projection with disk regions, is an invariant of the link.

We point out that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure eight knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial.

We establish a relation between the "large r" asymptotics of the TuraevViro invariants $TV_r $and the Gromov norm of 3manifolds. We show that for any orientable, compact 3manifold $M$, with (possibly empty) toroidal boundary, $logTVr(M)$ is bounded above by a function linear in $r$ and whose slope is a positive universal constant times the Gromov norm of $M$. The proof combines TQFT techniques, geometric decomposition theory of 3manifolds and analytical estimates of $6j$symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $logTVr(M)\geq B r$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3manifolds whose $SO(3)$ TuraevViro invariants grow exponentially. These constructions are essential for the results of article [3] where we make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations. We also study the behavior of the TuraevViro invariants under cutting and gluing of 3manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question ofmore »

We observe that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a $(2,q)$torus knot.