MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 3 qualifier

Just some small comments here. Yes, the question of whether or not Vassiliev invariants separate knots is still open.

One broader context for the question is to consider how Vassiliev invariants were first observed -- via the Vassiliev spectral sequence for the space of "long knots". These are knots of the form $\mathbb R \to \mathbb R^3$ which restrict to a standard (linear) inclusion outside of an interval, say $[-1,1]$.

There's a few standard ways in which the space of long knots can be turned into a topological monoid -- a "Moore loop space" construction which is pretty standard, or a "fat knots" construction from my "little cubes and long knots" paper. So this monoid operation is just the connect-sum operation, suitably jazzed-up to be strictly associative.

An observation for which I'm not sure if anyone has written up a complete proof yet is that the Goodwillie embedding calculus (an alternative approach to the Vassiliev spectral sequence, among other things) factors through the group-completion of the space of long knots. By this I mean, if K is the space of long knots, call the "group completion" the map $K \to \Omega BK$. $\Omega BK$ is where the "formal inverse" to a knot lives. The homotopy-type of $\Omega BK$ is computed in "little cubes and long knots". In particular, you can think of the Vassiliev spectral sequence as being an invariant of $\Omega BK$, not $K$. From the point of view of Vassiliev invariants this isn't such a major insight as $\pi_0 \Omega BK$ is the group-completion of the monoid $\pi_0 K$, so it's just a free abelian group on countably-many generators.

In particular, there are classes in $H^1 K$ for which there are no obvious finite-type approximations. $H_1 K$ has some torsion classes for which it's pretty unclear how to detect using the Vassiliev spectral sequence. Here is a mass of torsion computations for the homology of the long knot space.

edit: although "little cubes and long knots" describes the homotopy-type of the group completion of the space of long knots, it would be nice to have a more "concrete" embedding-space type description of the group completion. Mostovoy has an attempt here. I believe an idea like this should work and likely Mostovoy's idea works as well, but (again) this is something that needs to be written up for which there hasn't been enough time to start the project.
show/hide this revision's text 2 link to torsion computations

Just some small comments here. Yes, the question of whether or not Vassiliev invariants separate knots is still open.

One broader context for the question is to consider how Vassiliev invariants were first observed -- via the Vassiliev spectral sequence for the space of "long knots". These are knots of the form $\mathbb R \to \mathbb R^3$ which restrict to a standard (linear) inclusion outside of an interval, say $[-1,1]$.

There's a few standard ways in which the space of long knots can be turned into a topological monoid -- a "Moore loop space" construction which is pretty standard, or a "fat knots" construction from my "little cubes and long knots" paper. So this monoid operation is just the connect-sum operation, suitably jazzed-up to be strictly associative.

An observation for which I'm not sure if anyone has written up a complete proof yet is that the Goodwillie embedding calculus (an alternative approach to the Vassiliev spectral sequence, among other things) factors through the group-completion of the space of long knots. By this I mean, if K is the space of long knots, call the "group completion" the map $K \to \Omega BK$. $\Omega BK$ is where the "formal inverse" to a knot lives. The homotopy-type of $\Omega BK$ is computed in "little cubes and long knots"knots". In particular, you can think of the Vassiliev spectral sequence as being an invariant of $\Omega BK$, not $K$. From the point of view of Vassiliev invariants this isn't such a major insight as $\pi_0 \Omega BK$ is the group-completion of the monoid $\pi_0 K$, so it's just a free abelian group on countably-many generators.

In particular, there are classes in $H^1 K$ for which there are no obvious finite-type approximations. $H_1 K$ has some torsion classes for which it's pretty unclear how to detect using the Vassiliev spectral sequence. Here is a mass of torsion computations for the homology of the long knot space.

show/hide this revision's text 1

Just some small comments here. Yes, the question of whether or not Vassiliev invariants separate knots is still open.

One broader context for the question is to consider how Vassiliev invariants were first observed -- via the Vassiliev spectral sequence for the space of "long knots". These are knots of the form $\mathbb R \to \mathbb R^3$ which restrict to a standard (linear) inclusion outside of an interval, say $[-1,1]$.

There's a few standard ways in which the space of long knots can be turned into a topological monoid -- a "Moore loop space" construction which is pretty standard, or a "fat knots" construction from my "little cubes and long knots" paper. So this monoid operation is just the connect-sum operation, suitably jazzed-up to be strictly associative.

An observation for which I'm not sure if anyone has written up a complete proof yet is that the Goodwillie embedding calculus (an alternative approach to the Vassiliev spectral sequence, among other things) factors through the group-completion of the space of long knots. By this I mean, if K is the space of long knots, call the "group completion" the map $K \to \Omega BK$. $\Omega BK$ is where the "formal inverse" to a knot lives. The homotopy-type of $\Omega BK$ is computed in "little cubes and long knots". In particular, you can think of the Vassiliev spectral sequence as being an invariant of $\Omega BK$, not $K$. From the point of view of Vassiliev invariants this isn't such a major insight as $\pi_0 \Omega BK$ is the group-completion of the monoid $\pi_0 K$, so it's just a free abelian group on countably-many generators.

In particular, there are classes in $H^1 K$ for which there are no obvious finite-type approximations. $H_1 K$ has some torsion classes for which it's pretty unclear how to detect using the Vassiliev spectral sequence.