Topology of Morse flows with fixed points on the boundary of complete handlebody

Abstract

We studied topological properties of polar Morse flows on 3-dimensional handlebody, all fixed points of which lie on the boundary and have no separatrices connecting the saddle. We built an analog of Heegaard decomposition and $m$-diagram, which is complete topological invariant of flow. Equivalence of m-diagrams and flow we check using the distinguishing graphs. We found all possible distinguishing graphs with no more than 5 vertex.