#### Volume 17, issue 3 (2017)

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Embedding calculus knot invariants are of finite type

### Ryan Budney, James Conant, Robin Koytcheff and Dev Sinha

Algebraic & Geometric Topology 17 (2017) 1701–1742
##### 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-$\left(n-1\right)$ knot invariant. We compute the ${E}^{2}$–page in total degree zero for the spectral sequence converging to the components of this tower: it consists of $ℤ$–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 connected 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.

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##### Keywords
finite-type knot invariants, calculus of functors, embedding calculus, Taylor tower for the space of knots, configuration spaces, mapping space models, evaluation maps, stacking long knots, cosimplicial spaces, spectral sequences
##### Mathematical Subject Classification 2010
Primary: 55P65, 57M25
##### Publication
Received: 15 February 2016
Revised: 16 August 2016
Accepted: 19 September 2016
Published: 17 July 2017
##### Authors
 Ryan Budney Mathematics and Statistics University of Victoria PO Box 1700 STN CSC Victoria, BC V8W 2Y2 Canada James Conant Department of Mathematics University of Tennessee 227 Ayres Hall 1403 Circle Dr Knoxville, TN 37996 United States Robin Koytcheff Department of Mathematics and Statistics University of Massachusetts-Amherst Leaderless Graduate Research Tower Amherst, MA 01003 United States Dev Sinha Department of Mathematics University of Oregon Eugene, OR 97403 United States