Topology of two-connected graphs and homology of spaces of knots
Victor Anatolievich Vassiliev · Translations - American Mathematical Society/Translations · 1999
. We propose a new method of computing cohomology groups of spaces of knots in R n , n 3, based on the topology of configuration spaces and twoconnected graphs, and calculate all such classes of order 3: As a byproduct we define the higher indices, which invariants of knots in R 3 define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a certain finitedimensional configuration space and characterizes our class modulo the classes of smaller filtration. 1. Introduction The knots, i.e. smooth embeddings S 1 ! R n , n 3; form an open dense subset in the space K j C 1 (S 1 ; R n ). Its complement \\Sigma is the discriminant set, consisting of maps, having selfintersections or singularities. Any cohomology class fl 2 H i (K n \\Sigma) of the space of knots can be described as the linking number with an appropriate chain of codimension i + 1 in K lying ...