Embedding calculus knot invariants are of finite type

TL;DR

This paper demonstrates that embedding calculus knot invariants are finite type and explores their algebraic structure, providing evidence for their universality as finite-type invariants over integers.

Contribution

It establishes that the map from knots to the embedding calculus tower is a finite type invariant and develops a compatible group structure on the tower using advanced operad techniques.

Findings

The map is a finite type-(n-1) knot invariant.

Computed the second page of the spectral sequence for primitive chord diagrams.

Provided evidence that the tower is a universal finite-type invariant over integers.

Abstract

We show that the map on components from the space of classical long knots to the n-th stage of its Goodwillie-Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n-1) knot invariant. We also compute the second page in total degree zero for the spectral sequence converging to the components of this tower as Z-modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connect-sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps, and cosimplicial and subcubical diagrams.