Galois symmetries of knot spaces
Pedro Boavida de Brito, Geoffroy Horel
TL;DR
The paper reveals a Galois-type symmetry on knot spaces by constructing a Grothendieck-Teichmüller action at primes $p$ on the $p$-completed Little Disks operads and propagating this action to the Goodwillie-Weiss tower for knots. Leveraging these GT$_p$-symmetries and $p$-completion, the authors prove vanishing ranges for differentials in the $p$-localized spectral sequence, yielding computable information about knot spaces and their finite-type (Vassiliev) invariants. They show that the $(n+1)$-st Goodwillie-Weiss approximation is a $p$-local universal Vassiliev invariant of degree $\le n$ for $n\le p+1$, connecting embedding calculus with finite-type theory and stabilizing the Kontsevich-type rational splitting in a purely homotopy-theoretic way. In the homology setting, they obtain precise differential vanishing in the $p$-adic Bousfield-Kan spectral sequence and rational collapse results, contributing to a deeper understanding of the interplay between knot spaces, operadic formality, and Galois actions. Overall, the work links high-level operadic symmetries to concrete computational tools in knot theory via Goodwillie-Weiss calculus and $p$-localization.
Abstract
We exploit the Galois symmetries of the little disks operads to show that many differentials in the Goodwillie-Weiss spectral sequences approximating the homology and homotopy of knot spaces vanish at a prime $p$. Combined with recent results on the relationship between embedding calculus and finite-type theory, we deduce that the $(n+1)$-st Goodwillie-Weiss approximation is a $p$-local universal Vassiliev invariant of degree $\leq n$ for every $n \leq p + 1$.