Trajectory equivalence of optimal Morse flows on closed surfaces

Abstract

We consider optimal Morse flows on closed surfaces. Up to topological trajectory equivalence such flows are determined by marked chord diagrams. We present list all such diagrams for flows on nonorientable surfaces of genus at most 4 and indicate pairs of diagrams corresponding to the flows and their inverses.

Heegaard diagrams and optimal Morse flows on non-orientable 3-manifolds of genus 1 and genus $2$

“…Proof. A proof of this theorem in dimension 2 can be found in [4]. For arbitrary dimension greater than 1, it can be proved by the correspondence between Morse flows and Morse functions discussed in [16] and the well-known fact from Morse theory that any Morse function on a closed connected manifold with minimal number of critical points has one local minimum and one local maximum.…”

Topology of optimal flows with collective dynamics on closed orientable surfaces

“…We say that a flow from some class X (M ) of flows on a surface M is optimal if it has the least number of fixed points among all flows from X (M ). A Morse flow on a closed surface is optimal if and only if it has only one sink and one source [5]. Such a flow is also called a polar Morse flow.…”

“…Such a flow is also called a polar Morse flow. The topological structure of polar (optimal) Morse flows on closed surfaces was described in [3,10,4,5]. It is convenient to use chord diagrams as complete topological invariants of polar Morse flows.…”