http://arxiv.org/pdf/0909.0939 MTd2 spotted this paper back when it came out on 6 September and posted a reminder on another thread suggesting that we should discuss it. We should. It looks like a paper that is both important in the development of LQG and also exceptionally clear and instructive about somej basic things. I have to go out now but since MTd2 has not started a discussion thread for this one, I'll do it. Here's the abstract: Spin-Foams for All Loop Quantum Gravity Wojciech Kamiński, Marcin Kisielowski, Jerzy Lewandowski 23 pages, 8 figures (Submitted on 4 Sep 2009) "The simplicial framework of spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity. The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams. The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations). The notion of embedded spin-foam we introduce allows to consider knotting or linking spin-foam histories. The main tools are successfully generalized: the spin-foam vertex structure, the vertex amplitude, the Barrett-Crane as well as Engle-Pereira-Rovelli-Livine intertwiners. The correspondence between all the SU(2) intertwiners and the SU(2) x SU(2) EPRL intertwiners is proved to be 1-1 in the case of the Barbero-Immirzi parameter [tex]|\gamma|\ge 1[/tex]." Great paper! wish I could talk about it now, but I'm already late.