Related papers: Embedding calculus and Vassiliev spectral sequence
We develop a spectral approach to Sidorenko-type inequalities and apply it to establish sharp edge-spectral supersaturation results. Let $H$ be a bipartite graph with $v$ vertices and $e$ edges, where $v\le e$, and write $M(G)=2e(G)$. We…
We study invariant Seifert surfaces for strongly invertible knots, and prove that the gap between the equivariant genus (the minimum of the genera of invariant Seifert surfaces) of a strongly invertible knot and the (usual) genus of the…
The "fundamental theorem of Vassiliev invariants" says that every weight system can be integrated to a knot invariant. We discuss four different approaches to the proof of this theorem: a topological/combinatorial approach following M.…
Vassiliev's spectral sequence for long knots is discussed. Briefly speaking we study what happens if the strata of non-immersions are ignored. Various algebraic structures on the spectral sequence are introduced. General theorems about…
As it is well-known, all Vassiliev invariants of degree one of a knot $K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such invariants…
Recently, Sinha defined a spectral sequence approximating the (co)homology of the space of long knots in R^m modulo immersions, stemming from a cosimplicial structure on the compactified configuration spaces \`a la Kontsevich. We provide an…
This book is a detailed introduction to the theory of finite type (Vassiliev) knot invariants, with a stress on its combinatorial aspects. It is intended to serve both as a textbook for readers with no or little background in this area, and…
The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of…
On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature $\Theta_{f_g}(M)$, and squared $L^2$ norm of the mean curvature…
We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group.
We produce combinatorial formulas for invariants of smooth embeddings of $(2\ell-1)$-spheres into $\mathbb{R}^{3\ell}$ for $\ell\geq 2$. Furthermore, we obtain such a formula for the Haefliger invariant, which classifies smooth knots…
Lambrechts, Turchin and Voli\'c proved the Bousfield-Kan type rational homology spectral sequence associated to the $d$-th Kontsevich operad collapses at $E^2$-page if $d\geq 4$. The key of their proof is formality of the operad. In this…
We establish a pseudoisotopy result for embedding spaces in the line of that of Weiss and Williams for diffeomorphism groups. In other words, for $P\subset M$ a codimension at least three embedding, we describe the difference in a range of…
In the paper we describe complexes whose homologies are naturally isomorphic to the first term of the Vassiliev spectral sequence computing (co)homology of the spaces of long knots in R^d, d>=3. The first term of the Vassiliev spectral…
We construct a cubical CW-complex CK(M^3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M^3. We construct \bar{CK}(R^3) by attaching cells to CK(R^3) for every degenerate 1-singular and…
On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no…
We construct a spectral sequence relating the Khovanov homology of a strongly invertible knot to the annular Khovanov homologies of the two natural quotient knots. Using this spectral sequence, we re-prove that Khovanov homology…
Given a link $L$, Dowlin constructed a filtered complex inducing a spectral sequence with $E_2$-page isomorphic to the Khovanov homology $\overline{Kh}(L)$ and $E_\infty$-page isomorphic to the knot Floer homology $\widehat{HFK}(m(L))$ of…
We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound $\text{Ric}_{\infty} \geq K > 0$. Our main results show that if the equality holds, the manifold necessarily admits a…
This paper is a computation of the homotopy type of K, the space of long knots in R^3, the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopy class of long knot, and we `enumerate'…