Canonical Lyapunov Graphs and the Morse Polytope

In this article we will show that, in general, for each integral point $(\gamma_0, \ldots, \gamma_n)$ in the Morse polytope, $\mathcal{P}_{\kappa}(h_0, \ldots, h_n)$, one can associate an abstract Lyapunov graph $L(h_0, \ldots, h_n,\kappa)$ with $ntd$-labelling and realize a corresponding flow on $M^n$, where the Betti numbers of $M^n$ satisfy $\beta_j(M^n)= \beta_{n-j}(M^n)=\gamma_j$, for all $0