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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

We show that the map on components from the space of classical long knots to the nth 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-(n1) knot invariant. We compute the E2–page in total degree zero for the spectral sequence converging to the components of this tower: it consists of –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.

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
Received: 15 February 2016
Revised: 16 August 2016
Accepted: 19 September 2016
Published: 17 July 2017
Ryan Budney
Mathematics and Statistics
University of Victoria
PO Box 1700 STN CSC
Victoria, BC V8W 2Y2
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