On Quinn's Invariants of 2-dimensional CW-complexes

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Given a semisimple stable autonomous tensor category over a field K, to any group presentation with finite number of generators we associate an element Q(P)∈ K invariant under the Andrews-Curtis moves. We show that in fact, this is the same invariant as the one produced by the algorithm of Frank Quinn. The new definition allows us to present a relatively simple proof of the invariance and to evaluate Q(P) for some presentations. On the basis of some numerical calculations over different Gelfand-Kazhdan categories, we make a conjecture which relates the value of Q(P) for two different classes of presentations.