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