Statistics in 3d gravity from knots and links

A bstract In recent years, there has been remarkable progress in evaluating wormhole amplitudes in 3d Einstein gravity with negative cosmological constant and matching them to statistics of 2d CFT data. In this work, we compute non-perturbative Gaussian and non-Gaussian gravitational contributions to the OPE statistics using a framework that can systematically generate a class of such non-perturbative effects — Fragmentation of knots and links by Wilson lines . We illustrate this idea by constructing multi-boundary wormholes from fragmentation diagrams of prime knots and links with upto five crossings. We discuss fragmentations of hyperbolic knots and links like the figure-eight knot, the three-twist knot and the Whitehead link; and non-hyperbolic ones like the Hopf link, the trefoil knot, the Solomon’s knot and the Cinquefoil knot. Using Virasoro TQFT, we show how the partition functions on wormholes constructed from different fragmentations of the same knot or link are closely related. Using these fragmentations, we compute gravitational contributions to the variance, a two-point non-Gaussianity, two structures of four-point non-Gaussianities called the ‘pillow contraction’ and the ‘6 j -contraction’, and some six-point non-Gaussianities. We also check the consistency of some of these non-Gaussianities with the extended Gaussian ensemble of OPE data that incorporates the Gaussian corrections to the variance from knots. Finally, we comment on how fragmented knots contribute to the sum over geometries in 3d gravity through their relation to rational tangles.