Counting fixed points free vector fields on ${\mathbb{B}}^2$

Abstract

The number of diagrams of stationary points free vector fields in the 2-disk ${\mathbb{B}}^2$ is counted in the article. It is shown that the number of such diagrams with $2k$ exceptional points on the boundary ${\mathbb{S}}^1$ equals $3^{k-2}(C_k+2C_{k-1})$, where $C_k$ is the corresponding Catalan number. An algorithm for finding all such diagrams is discussed.

14 pages, 11 figures