I will explain how Feynman diagrams arise in pure algebra: how the computation of compositions of maps of a certain natural class, from one polynomial ring into another, naturally leads to a certain composition operation of quadratics and to Feynman diagrams.

I will also explain, with very little detail, how this is used in the construction of some very well-behaved poly-time computable knot polynomials, and then with better detail, why I care about having such invariants.

Joint work with Roland van der Veen.

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