First-order invariants and cohomology of spaces of embeddings of self-intersecting curves in~$\mathbb R^n$

V. A. Vassiliev

Izvestiya. Mathematics , Tome 69 (2005) no. 5, pp. 865-912

Résumé

We study the cohomology of the space of generic immersions $\mathbb R^1\to\mathbb R^n$, $n\geqslant3$, with a fixed set of transversal self-intersections. In particular, we study isotopy invariants of such immersions when $n=3$, calculate the lower cohomology groups of this space for $n>3$, and define and calculate the groups of first-order invariants of such immersions for $n=3$. We investigate the representability of these invariants by rational combinatorial formulae that generalize the classical formula for the linking number of two curves in $\mathbb R^3$. We prove the existence of such combinatorial formulae with half-integer coefficients and construct the topological obstruction to their integrality. As a corollary, it is proved that one of the basic 4th order knot invariants cannot be represented by an integral Polyak–Viro formula. The structure of the cohomology groups under investigation depends on the existence of a planar curve with a given self-intersection type. On the other hand, one can use the self-intersection type to construct automatically a chain complex calculating these cohomology groups. This gives a simple homological criterion for the existence of such a planar curve.