On height isotopy classes of embeddings in the plane of a Morse function of a circle

Abstract

Let $\mathrm{SM}_{2n}(S^1,\mathbb {R})$ be a set of stable Morse functions of an oriented circle such that the number of singular points is $2n\in \mathbb {N}$ and the order of singular values satisfies the particular condition. For an orthogonal projection $\pi :\mathbb {R}^2\rightarrow \mathbb {R}$, let ${\tilde{f}}_0$ and ${\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2$ be embedding lifts of f. If there is an ambient isotopy $\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2$ $(t\in [0,1])$ such that ${\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1$ and $\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1$, we say that ${\tilde{f}}_0$ and ${\tilde{f}}_1$ are height isotopic. We define a function $I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}$ as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals $2^{n-1}$.