Coloured Jones invariants of links and the Volume Conjecture

Abstract

We extend the definition of the coloured Jones polynomials to framed links and trivalent graphs in S 3 #k(S 2 × S 1 ) using a state-sum formulation based on Turaev's shadows. Then, we prove that the natural extension of the Volume Conjecture holds true for an infinite family of hyperbolic links.

6j–symbols, hyperbolic structures and the volume conjecture

“…It is well known that if L is a knot in S 3 then the order is at least 1 for each n since J n is divisible by OEn. More in general, one could expect the above limit to be related to the presence of essential spheres in the link complement; for instance, in [4], we proved that if L is a fundamental shadow link in # k S 2 S 1 , the leading order is 1 k for every n.…”

“…In [4] we showed how to extend the definition of colored Jones invariants to the case of links in #khS 2 S 1 i: in this case the resulting invariant has values in ‫.ޑ‬q…”

“…In [4], we showed that, when N is S 3 or a connected sum of S 2 S 1 , for r big enough, the set of admissible states stabilizes to a finite set so that the Reshetikhin-Turaev invariants can be seen as evaluations of a single rational function depending only the three …”

“…Proof We limit ourselves to sketch the key-idea of the proof: we refer to [4] for the details. If L is a fundamental shadow link, one can find a shadow P of L in which each 2-dimensional stratum contains a component of @P D L. Moreover, P is a simple polyhedron and contains a codimension 2 singularity per each vertex of the graph G used to describe L (see Definition 2.6).…”

“…In [4], we proposed an "extension" of the VC to include the case of links in S 3 or in connected sums of copies of S 2 S 1 : we defined colored Jones invariants taking values in the ring of rational functions for links in these more general ambient manifolds and proved the conjecture for the infinite family of "fundamental shadow links" (called universal hyperbolic links in the paper). These links were already studied by myself and D P Thurston [5] because of their remarkable geometric properties.…”