Abstract
The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time.
The invariants are parameterised by an integer \(r \ge 3\). We resolve the question of complexity for \(r=3\) and \(r=4\), giving simple proofs that computing Turaev-Viro invariants for \(r=3\) is polynomial time, but for \(r=4\) is #P-hard. Moreover, we give an explicit fixed-parameter tractable algorithm for arbitrary \(r\), and show through concrete implementation and experimentation that this algorithm is practical—and indeed preferable—to the prior state of the art for real computation.
A full version of this article is available at arXiv:1503.04099.
J. Spreer—Supported by the Australian Research Council (projects DP1094516, DP140104246).
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Burton, B.A., Maria, C., Spreer, J. (2015). Algorithms and Complexity for Turaev-Viro Invariants. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_23
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DOI: https://doi.org/10.1007/978-3-662-47672-7_23
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