Title:

Vector fields on surfaces

We consider "minimal" vector fields on a surface S with genus g. These are nondegenerate vector fields with the minimal number of vanishing points that satisfy a set of technical conditions to exclude pathological cases. We show that a minimal vector field gives rise to a directed graph with 2g  2 vertices such that each vertex has two edges entering and leaving it, a "dual" pair of circuit decompositions of equal size and a function that pairs up the circuits of this dual pair. Conversely, we show that given such a graph with a pair of circuit decompositions and such a function we can construct a unique minimal vector field. This correspondence enables us to classify these vector fields. The proof of the correspondence result requires several invariants, one each from graph theory and the topology of the surface. These invariants are, respectively, the directed graph G formed by the noncompact flowlines of the vector field and a neighbourhood of this graph. Invariants arising from the homology of the pair (S, V) are also discussed, where V is the set of vertices of the directed graph G. Further, we show how to construct all possible minimal vector fields from a given graph provided the graph satisfies certain natural properties and give an algorithm that identifies which circuit decompositions have a suitable dual. We obtain some new results on the Martin polynomial, a skeintype polynomial of graphs first identified by P. Martin (1977). Some other combinatorial results concerning polynomials and graphs are proved.
