The embedding-calculus universality conjecture for classical knots

Let $I=[0,1]$, let $\Knots(I^3)=\Emb_\partial(I,I^3)$ be the space of long knots, and write $\KK(I^3):=\pi_0\Knots(I^3)$. For each $n\geq 1$, let $\pT_n(I^3)$ be the $n$th stage of the embedding-calculus tower, let $\ev_n\colon\Knots(I^3)\to\pT_n(I^3)$ be its evaluation map, and let $\sim_n$ be the finite-type equivalence relation on $\KK(I^3)$ defined by claspers or gropes. A universal additive Vassiliev invariant of type $\leq n-1$ over $\mathbb Z$ is a monoid homomorphism factoring through $\KK(I^3)/\sim_n$ with induced target group isomorphic to that quotient. Embedding-calculus universality conjecture. For each $n\geq 1$, the map

$$\pi_0\ev_n\colon\KK(I^3)\to\pi_0\pT_n(I^3)$$

is a universal additive Vassiliev invariant of type $\leq n-1$ over $\mathbb Z$: it is a monoid homomorphism which factors through $\KK(I^3)/\sim_n$, and the induced map

$$\KK(I^3)/\sim_n\xrightarrow{\cong}\pi_0\pT_n(I^3)$$

is an isomorphism of groups. The conjecture relates Vassiliev's finite-type theory to the Goodwillie–Weiss embedding calculus for long knots. The parser records it as resolved: work of Volić, Turchin and Conant established agreement of the relevant graph complexes, and part of the conjecture was subsequently confirmed by work cited as BCKS.