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 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-(n − 1) knot invariant. We compute the E 2 -page in total degree zero for the spectral sequence converging to the components of this tower, identifying it 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.

Homotopy Bott–Taubes integrals and the Taylor tower for spaces of knots and links

“…While this matches Volić's result that T n K factors all rational type-n 2 knot invariants, Conjecture 1.1 of [3] is that T n K factors all type-(n − 1) knot invariants. (Our work in [2] provides some evidence for this conjecture). We might similarly expect to see the type-2 link invariant μ 123 at a stage (n 1 , n 2 , n 3 ) with n 1 + n 2 + n 3 = 3.…”

“…Thus the homotopy-theoretic BottTaubes integrals provide a way of factoring μ 123 through the Taylor tower for any stage as low as T (2,2,2) …”

“…We showed in [2] that each stage of the Taylor tower for the space of long knots is an H-space with an operation compatible with connect-sum of long knots. There we used a variant of the aligned maps model where the n-th stage is roughly a space of maps of n ∼ = C n I to the simplicial compactification C n I d of configuration space.…”

“…We use a "punctured links" model for the Taylor tower that has appeared in work of other authors, e.g., [21,23,26]. Ultimately, we deduce that μ 123 factors through the stage T (2,2,2) L 3 3 of the multi-tower. We could deduce this latter result via integration of forms, using Theorem 3.4, which produces all finite-type link invariants in the tower by integration.…”

“…Note that because of the blowups performed in constructing C n [M], any stratum indexed by a single set {S} has codimension one, regardless of the cardinality of S 2. We have omitted the so-called anomaly term in our formula for BT .…”