Zesting and the relative complexity of Reshetikhin-Turaev invariants

Abstract

We show that the computational complexity of Reshetikhin-Turaev invariants of simply colored links is preserved when their underlying ribbon fusion categories are related by the zesting construction. Zesting modifies an $A$-graded ribbon fusion category $\mathcal{C}$ with additional algebraic data $ζ$ to produce a new category $\mathcal{C}^ζ$ whose link invariants are known to differ from those of $\mathcal{C}$ by an invariant of $A$-colored links $\mathcal{J}_ζ$ depending only on $ζ$. Building on this understanding and on earlier work on quantum braid group representations under zesting, our result suggests how zesting contributes to the organization of (2+1)D topological quantum field theories and topological phases into complexity-theoretic hierarchies. To prove our main result we develop a local formalism analogous to the Reshetikhin-Turaev construction to compute \emph{tangle} invariants $\mathcal{J}_ζ(T)$, which leads to a polynomial time algorithm to compute invariants of links $\mathcal{J}_ζ(L)$. A byproduct of our construction is an identification (up to a sign) of the link invariants $\mathcal{J}_ζ(L)$ as rack cocycle invariants, which may be of independent interest. Our formalism also extends to define invariants of closed $3$-manifolds with $A$-structure and we obtain similar complexity results for homotopy quantum field theories built from $A$-modular fusion categories.