Is there any analogs of Vassiliev invariants in higher dimensions?

Question

http://www.artofproblemsolving.com/Forum/blog.php?u=55354&b=32193

Today I asked my TA for the knot theory class whether there exists any analogs for Vassilev invariants, like at least in $ (2,4) $ case (to me the question is how to define singularities properly?). Also I asked him whether there exists any basic classifications (I guess we still use something like triangular moves). He told me to ask the professor, but the professor is not accessible, so I ask in here. I know the question seemed primitive (definitely not a research type question), I just don't know where I can find the needed reference.

Another question I want to ask is whether there exists any good integral type invariants. I feel it is highly unlikely to find one, but I don't know why. I think one criterion such an integral has to satisfy is it can distinguish small knotted parts in comparison with unknot parts, and it should be able to distinguish images with its mirror images.

Answer 1

The best work I know of along these lines was by Rossi, a student of Cattaneo, which is explained here: http://www.math.uzh.ch/fileadmin/math/preprints/07-05.pdf

There are a number of interrelated views of Vassiliev theory. One approach proceeds from Chern-Simons theory and produces "configuration space integrals" which generalize the Gauss integral that Sammy mentions. (By the way, these for example do generalize to linked S^n's in S^{2n+1}). These integrals were first studied by Bar-Natan in his PhD thesis and then by Bott-Taubes, Kontsevich, D. Thurston, Cattaneo and his collaborators, Volic and others. In topological terms, one takes a knot, looks at the "induced map on configuration spaces" and then uses that map in de Rham theory (including some difficult vanishing arguments) to produce knot invariants. The theory is yet to be completely understood even in dimension three, so as one might expect Cattaneo and Rossi only get a few kinds of configuration space integrals to work out in higher dimensions, but it is a significant starting point.

Answer 2

Sorry for taking so long to respond. Theo's answer is fairly complete. I mentioned the question to Masahico who reminded me that there are results of Kanenobu and others about finite type invariants for ribbon knots. The general case is known to be difficult if not impossible.

I don't agree that a framed braided monoidal 2-category with duals is going to be difficult. But in general finding braided monoidal 2-categories with duals that give non-trivial invariants is a difficult enterprise. I asked one expert about this. He said there were lots, so I said, "Ok name one." See Kevin Walker's response to this question.

Also, before looking at our stuff, someone who wants to learn about knotted surfaces should look at Roseman's paper.

To find non-trivial invariants, whether or not they be finite-type, the best bet so far is the quandle-cocycle invariant.

Now, to return to an answer that the questioner may appreciate: the crossing points of a classical knot are isolated. The crossing points for surfaces are arcs and circles. Crossing changes are not local things. We don't even know if we can unknot by switching crossings between sheets, and sometimes it is hard to imagine how to push a crossing change through a triple point.

There is a lot to learn about knotted surfaces. Yeah sure, you can look at our books, but one should also read the elegant papers by Kanenobu, Satoh, and Hillman.

Answer 3

This is not a complete answer --- I hope that experts can give more information.

The question of whether there exist higher-dimensional finite-type invariants is a very good one. The story has not been completely worked out. A good place to start is the work by Carter and Saito and collaborators form the 90s. In particular, the classification for the ways that surfaces can intersect generically is essentially known. Surfaces can intersect along a one-dimensional subspace in only one way --- locally the intersection is modeled on the intersection of two planes in R^3. But interesting things can happen at codimension-two, which I will describe in terms of local models (I'm going from memory that these are all there are):

• the triple intersection of the xy, yz, and zx plane.
• the ramified point above 0 in the Riemann surface for \sqrt{z}, or equivalently the tip of a cone glued to a sheet.

But in general there is a lot of the story of finite-type knot invariants that have not been told. In particular, I don't think we are anywhere close to a Kontsevich-type theorem. I mean, I'm not sure anyone has really said what the thing corresponding to the chord diagrams are.

In general, what should happen is the following. One should come up with a precise notion of "framed braided monoidal 2-category with duals", which allows for the possibility of nontrivial "associators" --- framed braided monoidal categories with duals are precisely the things that can interpret knots as numbers, and so "framed braided monoidal 2-category with duals" should be able to interpret 2-knots. Then one should say what the "symmetric" ones are, and describe the deformation problem for "framed braided monoidal 2-category with duals". In particular, the one-jet of a deformation will be some sort of "framed infinitesimally-braided symmetric monoidal 2-category with duals". Whatever these are, they should be precisely the type of algebraic gadgetry for interpreting "2-chord diagrams". Then the deformation (or "quantization") problem begins with the initial data of a framed infinitesimally-braided symmetric monoidal 2-category with duals and asks for a framed braided monoidal 2-category with duals over C[[h]] for which the original framed infinitesimally-braided symmetric monoidal 2-category is the one-jet of the deformation (the C[h]/h^2 part). The answer should be that there is some sort of "higher Drinfel'd associator" that solves this deformation problem coherently.

I think the one-dimensional version of this story was first told in this language by Cartier in roughly 1993. It is explained well in the book by Kassel, Rosso, and Turaev from 1999. Bits of the two-dimensional story are also in Baez's HDA4, although they do not consider nontrivial associators, and the one-dimensional story makes it clear that such associators are important. Schommer-Pries's thesis has some of the story in it as well --- he does a good job of recalling the Carter&Saito work, so it's a good choice if you only want to read only one thing, although I don't think he actually discusses finite-type things.

Answer 4

First, let me recall briefly Vassiliev's theory (the way I understand it). I will skip many important details on the way.

Vassiliev's invariants first appeared as certain cohomology classes of the space of all knots in the 3-space. The strategy is as follows: instead of the space of all knots, which is infinite-dimensional, consider the space $V_d$ of maps from $\mathbf{R}$ to $\mathbf{R}^3$ given by polynomials of some fixed degree $d$ behaving in a prescribed way at infinity. This is a finite dimensional space and non-knots form a hypersurface $\Sigma_d$ in it. Using the Alexander duality one can reduce the computation of the cohomology of the complement of $\Sigma_d$ in $V_d$ to computing the Borel-Moore homology of $\Sigma_d$. [Note: there is a price to pay for replacing an open set with its complement -- we loose the information of how $\Sigma_d$ lies in $V_d$.] The Borel-Moore homology of $\Sigma_d$ can be computed by constructing a suitable semi-simplical resolution of $\Sigma_d$ (i.e. a semi-simplicial space with the geometric realization properly homotopy equivalent to $\Sigma_d$).

The geometric realization of a semi-simplicial space admits a natural filtration, which gives a spectral sequence converging to the Borel-Moore homology of $\Sigma_d$. There is a way to embed $V_d$ into some $V_{d'},d'>d$ so that knots are mapped to knots and unknots to unknots. Moreover, spectral sequences for $d'$ and $d$ are mapped to one another. This corresponds to the restriction map in cohomology, but constructing the maps on the level of spectral sequences is non-trivial. As $d$ goes to $\infty$ a part of the $E^1$ term of the spectral sequence stabilizes. It is a difficult theorem (Kontsevich-Vassiliev) that (at least over the rationals) the second and higher differentials vanish on the stable part and that if an element in the stable part is killed, it is killed in the first sheet by some stable element for all sufficiently large $d$.

So letting $d$ go to $\infty$ we get a vector subspace (in fact, a subring) in the cohomology of the space of all knots (equipped with the Whitney topology), together with a filtration on it. Vassiliev's conjecture says that this subspace is in fact the whole of the cohomology. By taking the $H^0$ part we get Vassiliev invariants and the Vassiliev filtration on them.

The most non-trivial part of the above is the actual construction of the semi-simplicial resolution of $\Sigma_d$ (since this gives a spectral sequence that collapses very fast).

The same strategy can be applied in many other cases, such as spaces of smooth functions without complicated singularities, spaces of mappings from $m$-dimensional CW-complexes to $m-1$ connected ones, spaces of smooth projective hypersurfaces, classical Lie groups etc. It can also be applied to spaces of knots in 3-manifolds other than $\mathbf{R}^3$ and in higher-dimensional manifolds. For $\mathbf{R}^n,n>3$ this gives a complete description of the cohomology: it is isomorphic to the cohomology of the Hochschild complex of the Poisson operad for $n$ odd and of the Gerstenhaber operad for $n$ even, as shown by V. Turchin arXiv:math/0010017. The case $n=3$ is much trickier. Some of these applications can be considered higher dimensional analogs of Vassiliev knot invariants.

Most of the above is described in detail in "Complements of the Discriminants of Smooth Maps" by Vassiliev. A brief synopsis can be found in Vassiliev's ICM 1994 talk.

Answer 5

This is a nice question. There is actually quite a bit of work which has been done along these lines, although we are a very long way from having a good understanding of how a theory of finite-type invariants should work for higher-dimensional knots.
Building on work of Habiro and Shima, Tadayuki Watanabe has pushed the idea of finite-type invariants of (ribbon) n-knots furthest, I believe, using higher-dimensional analogues of claspers. His theory is already quite impressive, and he can recover known K-theoretical calculations of characteristic classes of unknots from his formulae, and the connection with configuration space integrals is quite explicit. References:
On Kontsevich’s characteristic classes for smooth 5- and 7-dimensional homology sphere bundles math/0610292.
Configuration space integral for long n-knots, the Alexander polynomial and knot space cohomology math/0609742.
Clasper-moves among ribbon 2-knots characterizing their finite type invariants Journal of Knot Theory and Its Ramifications, 2006; 15 (9) 1163-1200

The other people working on this, as mentioned by Dev Sinha, are Cattaneo and Rossi
(Wilson surfaces and higher dimensional knot invariants, Comm. Math. Phys. 256 (2005) 513-537)
Cattaneo, Cotta-Ramasino, Longoni (Configuration spaces and Vassiliev classes in any dimension) Alg. Geom. Topol. 2 (2002) no.39 949-1000

Configuration space integrals (including self-linking integrals as the simplest example) for 2-knots were first studied I think by R. Bott, who found a CFI invariant for 2-knots.
Configuration spaces and embedding invariants, Turkish J. Math; 20(1) (1996) 1-17.

In another direction, Greg Kuperberg has a version of the Gauss integral which works to compute the linking number of two closed submanifolds of Sⁿ.
From the Mahler conjecture to Gauss linking forms, math/0610904.
DeTurck and Gluck have done further work in this direction. Furthermore, there is
Clayton Shonkwiler, David Shea Vela-Vick (Higher-dimensional linking integrals) math/0801.4022.
One of the basic properties of the Gauss integral is that integrand is invariant under orientation-preserving isometries of Euclidean space, which is important in geometric applications. They find a linking integral formula in higher dimensions which shares this property.

Answer 6

Gauss defined the linking number of two closed curves $\gamma_i: S^1 \to \mathbb{R}^3$ with an integral.

$L(\gamma_1, \gamma_2) = \frac{1}{4\pi} \oint_{\gamma_1}\oint_{\gamma_2} \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2)$.

In modern terms, this measures the degree of the Gauss map $\Gamma(\gamma_1,\gamma_2): S^1 \times S^1 \to S^2$.

See http://en.wikipedia.org/wiki/Linking_number.