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To better understand and compute knot invariants from Chern-Simons theory we introduce two new categories. The first is the category of rec-tangles, whose morphisms are quartic graphs on framed surfaces with some distinguished bands known as rectangles. The rectangles in one morphism may be identified with the thickened edges in the next to produce a composition that is similar to surgery on tangles in thickened surfaces. The topological significance of rec-tangles is that they allow us to deal locally with both knots and the natural operations on them. Algebraically speaking the rec-tangles provide a convenient graphical calculus for working in a quasi-triangular Hopf algebra (quantum group).

Morphisms in the second category are formal series which take the form of Gaussians, i.e. exponentials of quadratics in pairs of dual variables and composition is by contraction. Their significance is to compute efficiently in suitable PBW algebras (quantum groups again) by staying in a tiny fraction where the knot invariants actually take their values. Finally our invariants will be functors from rec-tangles to Gaussians and we illustrate this approach by computing the n-loop sl_2 invariant in polynomial time. This is joint work in progress with Dror Bar-Natan, a handout and additional material will be available at http://www.rolandvdv.nl/Talks/Davis1904/