Papers

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 $\zeta$ to produce a new category $\mathcal{C}^{\zeta}$ whose link invariants are known to differ from those of $\mathcal{C}$ by an invariant of $A$-colored links $\mathcal{J}_{\zeta}$ depending only on $\zeta$. 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 tangle invariants $\mathcal{J}_{\zeta}(T)$, which leads to a polynomial time algorithm to compute invariants of links $\mathcal{J}_{\zeta}(L)$. A byproduct of our construction is an identification (up to a sign) of the link invariants $\mathcal{J}_{\zeta}(L)$ 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.

We prove that every smooth two-component split sphere link $L\sqcup R\subset S^4$ admits infinitely many smooth splitting $3$-spheres that are topologically non-isotopic. This generalizes a theorem of Tatsuoka from the two-component sphere unlink to split links with arbitrarily knotted sphere components. In the course of the proof, we establish a general sufficient condition under which a connected sum of smooth $4$-manifolds admits infinitely many topologically non-isotopic splitting $3$-spheres. This criterion may be of independent interest; in particular, it applies to all previously known examples of nonuniqueness for splitting $3$-spheres of positive-genus surface links.

We study the expected number of fixed points of a random permutation obtained via a word map, with surface group constraints imposed. For stable irreducible characters $\chi$ of the symmetric group $S_{n}$ and with $R_{g}=[a_{1},b_{1}]\dots[a_{g},b_{g}]$ and $w\in F_{2g}$, we compute $\mathbb{E}_{S_{n}^{2g}}\left[\chi\left(R_{g}(h)\right)\#\mathrm{fix}\left(w(h)\right)\right]$. We show that, if $w$ is a shortest representative for the conjugacy class of $\gamma\in\Gamma_{g}=\left\langle a_{1},b_{1},\dots,a_{g},b_{g}:R_{g}\right\rangle$, then this expectation is $O\left(1/\dim\chi\right)$. As an application, we recover a boundedness statement of Magee–Puder on the large $n$ limit of the expected number of fixed points of $\phi_{n}(\gamma)$, where $\gamma\in\Gamma_{g}$ is fixed and $\phi_{n}\in\hom\left(\Gamma_{g},S_{n}\right)$ is chosen uniformly at random.

We determine the minimum hyperbolic area of Teichmuller curves arising from holomorphic quadratic differentials on closed Riemann surfaces of genus two. The minimum is 3\pi/5, and it is attained precisely by quadratic differentials q=\omega^2 for which the translation surface (X,\omega) lies in the GL_2^+(R)-orbit of the double-pentagon translation surface. Equivalently, the extremal projective Veech group is the triangle group \Delta(2,5,\infty). The proof combines a small-area classification of noncompact hyperbolic orbifolds with a derivative-preserving affine descent construction for nonsquare quadratic differentials. The three possible nonsquare zero patterns are then excluded by arithmetic, marked-point, and covering obstructions.

Given a Riemann surface {\Sigma}, let {\Gamma} {\subseteq} Mod({\Sigma}) denote the monodromy subgroup of a family of complex curves homeomorphic to {\Sigma}, arising from a sufficently ample Lefschetz pencil. We establish that the group {\Gamma} acts with Zariski dense orbits or ergodically on certain character varieties for {\pi_1}({\Sigma}). This answers a version of a Conjecture of Katzarkov, Pantev, and Simpson appearing in [KPS03].