Embedding calculus knot invariants are of finite type

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.

Revised maps to the infinitesimal mapping space model in Sections 3 and 4 and analysis of cubical diagrams in Section 5. Minor expository and organizational changes throughout. Now 28 pages, 4 figures

References in corpus (1)

• Long knots and maps between operads

Cited by in corpus (7)

• Spaces of smooth embeddings and configuration categories
• Galois symmetries of knot spaces
• Embedding calculus and grope cobordism of knots
• Operadic actions on long knots and 2-string links
• An Application of the $h$-principle to Manifold Calculus
• Goodwillie's cosimplicial model for the space of long knots and its applications
• Graphing, homotopy groups of spheres, and spaces of long links and knots