Algorithms and Complexity for Turaev-Viro Invariants

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

$$r \ge 3$$

r

≥

3

. We resolve the question of complexity for

$$r=3$$

r

=

3

and

$$r=4$$

r

=

4

, giving simple proofs that computing Turaev-Viro invariants for

$$r=3$$

r

=

3

is polynomial time, but for

$$r=4$$

r

=

4

is #P-hard. Moreover, we give an explicit fixed-parameter tractable algorithm for arbitrary

$$r$$

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.