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1.9 Isotopy Types of Harnack $ M$-Curves

Harnack's construction of $ M$-curves in [#!35s!#] differs from the construction in the proof of Theorem 1.6.B in that a conic, rather than a curve of degree 5, is used as the original curve. Figure 9 shows that the $ M$-curves of degree $ \le 5$ which are used in Harnack's construction [#!35s!#]. For $ m\ge 6$ Harnack's construction gives $ M$-curves with the same isotopy types as in the construction in Section 1.6.

Figure 9:
\begin{figure}\centerline{\epsffile{f9s.eps}}\end{figure}

In these constructions one obtains different isotopy types of $ M$-curves depending on the choice of auxiliary curves (more precisely, depending on the relative location of the intersections $ \mathbb{R}B_m\cap \mathbb{R}L)$. Recall that in order to obtain $ M$-curves it is necessary for the intersection $ \mathbb{R}B_m\cap \mathbb{R}L$ to consist of $ m$ points and lie in a single component of the set $ \mathbb{R}L\smallsetminus \mathbb{R}B_{m-1}$, where for odd $ m$ this component must contain $ \mathbb{R}B_{m-2}\cap \mathbb{R}L$. It is easy to see that the isotopy type of the resulting $ M$-curve of degree $ m$ depends only on the choice of the components of $ \mathbb{R}L\smallsetminus \mathbb{R}B_{r-1}$ for even $ r<m$ where the intersections $ \mathbb{R}L\cap \mathbb{R}B_r$ are to be found. If we take the components containing $ \mathbb{R}L\cap \mathbb{R}B_{r-2}$ for even $ r$ as well, then the degree $ m$ $ M$-curve obtained from the construction has isotopy type $ \langle J\amalg (m^2-3m+2)/2\rangle$ for odd $ m$ and $ \langle (3m^2-6m)/8\amalg 1\langle (m^2-6m+8)/8\rangle\rangle$ for even $ m$. In Table 2 we have listed the isotopy types of $ M$-curves of degree $ \le 10$ which one obtains from Harnack's construction using all possible $ B_m$.


Table 2:
$ m$ Isotopy types of the Harnack $ M$-curves of degree $ m$
2 $ \langle 1\rangle$
3 $ \langle J\amalg 1\rangle$
4 $ \langle 4\rangle$
5 $ \langle J\amalg 6\rangle$
6 $ \langle 9\amalg 1\langle
1\rangle\rangle$
7 $ \langle J\amalg 15\rangle\qquad
\langle J\amalg 13\amalg 1 \langle 1\rangle\rangle$
8 $ \langle 18\amalg 1\langle 3\rangle\rangle\qquad\qquad\qquad
\langle 17\amalg 1\langle 1\rangle \amalg 1\langle 2\rangle\rangle$
9 $ \langle J\amalg 28\rangle\qquad
\langle J\amalg 24\amalg 1\langle 3\rangle\ran...
...qquad
\langle J\amalg 23\amalg 1\langle 1\rangle\amalg 1\langle
2\rangle\rangle$
10 $ \langle 30\amalg 1\langle 6\rangle\rangle \qquad
\langle 29\amalg 2\langle 3\r...
...malg 1\langle 1\rangle \amalg 1\langle 2\rangle
\amalg 1\langle 3\rangle\rangle$

In conclusion, we mention two curious properties of Harnack $ M$-curves, for which the reader can easily furnish a proof.

Figure 10: Construction of even degree curves by Hilbert's method. Degrees 4 and 6.
\begin{figure}\centerline{\epsffile{f10s.eps}}\end{figure}

Figure 11: Construction of odd degree curves by Hilbert's method. Degrees 3 and 5.
\begin{figure}\centerline{\epsffile{f11s.eps}}\end{figure}

1.9.A   The depth of a nest in a Harnack $ M$-curve is at most 2.

1.9.B   Any Harnack $ M$-curve of even degree $ m$ has $ (3m^2-6m+8) /8$ outer ovals and $ (m^2-6m+8)/8$ inner ovals.


next up previous
Next: 1.10 Hilbert Curves Up: 1 Early Study of Previous: 1.8 End of the
Oleg Viro 2000-12-29