A classification of embeddings of punctured n-manifolds into R^{2n-1}

Dmitry Tonkonog

Moscow State University

Abstract

The talk will be on the Knotting Problem: for an n-dimensional manifold N and a number m describe the set E^m N of isotopy classes of embeddings of N into Euclidean space R^m. For a recent survey on the topic see arXiv:0604045 by A. Skopenkov.

For example, if N is a circle and m=3, we get the problem of classifying classical knots in R^3, which is recognized to be very hard. There are invariants of classical knots (such as the Jones polynomial) which are defined in simple geometric terms and can be calculated algorithmically. However, such invariants are /incomplete/, meaning that two non-isotopic knots can have the same invariant.

The Knotting Problem can become less complicated in higher dimensions. For example,

E^m N=0 for m >= 2n+1 and E^2n N=H_1(N;Z) or H_1(N;Z_2), depending on the parity of n for n=dim N>=4.

Thus the `first' nontrivial case is m=2n-1. It would be interesting to have some /complete/ geometric invariant of embeddings of N into R^{2n-1}.

In the talk, I will define such invariant, but for a slightly simpler problem. Namely, I will deal with embeddings of N_0 into R^{2n-1} instead, where N_0 is the punctured manifold obtained by cutting out an open n-ball from a closed manifold N. The invariant is defined using linking numbers.

If time allows, I will state a conjecture about a complete invariant of embeddings of a closed manifold N into R^{2n-1}.

The talk will mainly be based on author's preprint arXiv:1003.3029.