Danica Kosanović

We show that embedding calculus invariants $ev_n$ are surjective for long knots in an arbitrary 3-manifold. This solves some remaining open cases of Goodwillie–Klein–Weiss connectivity estimates, and at the same time confirms one half of the conjecture that for classical knots $ev_n$ are universal additive Vassiliev invariants over the integers. In addition, we give a sufficient condition for this conjecture to hold over a coefficient group, which is by recent results of Boavida de Brito and Horel fulfilled for the rationals and for the $p$-adic integers in a range. Therefore, embedding calculus invariants are strictly more powerful than the Kontsevich integral.

Furthermore, our work shows they are more computable as well. Namely, the main theorem computes the first possibly non-vanishing invariant $ev_n$ of a knot which is grope cobordant to the unknot to be precisely equal to the equivalence class of the underlying decorated tree of the grope in the associated graph complex. Actually, our techniques apply beyond dimension $3$, offering a description of the layers in embedding calculus for long knots in a manifold of any dimension, and suggesting that certain generalised gropes realise the corresponding graph complex classes.

Given for any $1\leq k\leq d$ a knot $s\colon {S}^{k-1}\hookrightarrow\partial M$ with a framed dual sphere $G\colon {S}^{d-k}\hookrightarrow\partial M$, we show that the space of neat embeddings $ {D}^k\hookrightarrow M^d$ with boundary $s$ is homotopy equivalent to the loop space of neatly embedded $(k-1)$-disks in the manifold obtained from $M$ by attaching a handle to $G$. This increase in the codimension significantly simplifies the study of homotopy groups of such embedding spaces, and is of interest also in low-dimensional topology.

In particular, we find a tight connection of this space-level result with recently given 4-dimensional constructions and conclude that an invariant going back to Dax gives a complete isotopy classification of homotopic disks in a 4-manifold. Moreover, we construct a group structure on the set of isotopy classes of all disks with boundary $s$ and show that it is usually neither abelian nor finitely generated and almost entirely injects into the mapping class group of $M$. We recover all previous results for 2-spheres in 4-manifolds and prove that in this setting the Dax invariant reduces to the Freedman--Quinn invariant.

We compute in many classes of examples the first potentially interesting homotopy group of the space of embeddings of either an arc or a circle into a manifold $M$ of dimension $d\geq4$. In particular, if $M$ is a simply connected 4-manifold the fundamental group of both of these embedding spaces is isomorphic to the second homology group of $M$, answering a question posed by Arone and Szymik. The case $d=3$ gives isotopy invariants of knots in a 3-manifold, that are universal of Vassiliev type $\leq1$, and reduce to Schneiderman's concordance invariant.

Finite type invariants (often called Gusarov-Vassiliev, or just Vassiliev, invariants) give a certain filtration on the set of all invariants by their type. A dual point of view is, however, more geometric: there is a filtration on the monoid of knots itself, which arises by looking at a certain sequence of n-equivalence relations on knots. Then the n-th term of the filtration is comprised of knots which are n-equivalent to the unknot.

For example, two knots are 1-equivalent if they can be related by a sequence of crossing changes. This means that the first term in the filtration is equal to the whole monoid of knots! To get an idea about 2-equivalence, take a look at the operation on the left - grab some three strands of a knot and connect-sum them with the Borromean rings.

Embedding calculus of Goodwillie and Weiss is another homotopy-theoretic approach to spaces of embeddings. When applied to the embedding functor of long knots $\mathcal{K}$ in the 3-space it yields a tower of spaces $T_n$ together with evaluation maps $ev_n\colon K\to T_n$. These spaces turn out to be very interesting. For example, they can be shown to be double loop spaces of the mapping spaces between some (truncated) operads. Hence, their components form an abelian group and the evaluation map from knots gives a map on $\pi_0$ which turns out to be a finite type invariant! It is conjectured to be universal such, in other words, the group of knots modulo relation of n-equivalence is isomorphic to $\pi_0T_n$.

Therefore, the two stories should not be so separate after all. One unifying perspective is that of gropes. Namely, the trivalent vertices appearing in the diagrams for finite type theory (originating in quantum Chern-Simons theory) correspond to the Borromean rings, and the isotopy depicted below hints at how this in turn relates to gropes. In the very last picture we clearly see a genus one surface with one boundary component emerging. This will represent the bottom stage of a grope.