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We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the W -matrix-the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the W -matrix and the set of punctured S-matrices are strictly beyond the modular data (S, T ). Whether or not the triple (S, T, W ) constitutes a complete invariant of modular categories remains an open question.

ABSTRACT. We provide an elementary introduction to topological quantum computation based on the Jones representation of the braid group. We first cover the Burau representation and Alexander polynomial. Then we discuss the Jones representation and Jones polynomial and their application to anyonic quantum computation. Finally we outline the approximation of the Jones polynomial and explicit localizations of braid group representations.

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Following Manin's approach to renormalization in the theory of computation, we investigate Dyson-Schwinger equations on Hopf algebras, operads and properads of flow charts, as a way of encoding self-similarity structures in the theory of algorithms computing primitive and partial recursive functions and in the halting problem.