Classification of flexible M-curves of degree 8 up to isotopy

S.Yu. Orevkov

Abstract:

We complete the classification of real pseudo-holomorphic M-curves on RP² of degree 8 up to isotopy. All real schemes which were known not to be algebraically unrealizable (due to results of Rohlin, Fiedler, Viro, Shustin) neither are pseudo-holomorphically  realizable. Here we exclude  1 U 1<3> U 1<16>  and  1 U 1<13> U 1<6>  and show that all the 7 remaining real schemes, namely   1 U 1<1> U 1<18> ,  1 U 1<4> U 1<15> , 1 U 1<7> U 1<12> , 1 U 1<9> U 1<10> , 4 U 1<2 U 1<14>> , 14 U 1<2 U 1<4>> , and  7 U 1<2 U 1<11>>  are realizable as the set of real points of a real pseudo-holomorphic M-curve of degree 8.

A complete list of all isotopy types realizable by pseudo-holomorphic M-curves of degree 8 is given in the paper.

The method of restriction: a study of the braid defined by a curve and a pencil of lines (this method was used in a series of my previos papers) combined with the following three new tools:

   1. Periodicity (with respect to k) of Tristram signatures of braid of the form $b \sigma_i^k \sigma_j^{-k}$.

   2. Degeneration of a curve in the class of pseudo-holomorphic curves using one pencil of lines followed by a consideration of the braid coming from another pecil of lines

   3. Analogue of Fox-Milnor theorem for the Alexander polynomial of a link of ribbon Euler characteristc one.

In an appendix we present computer programs (to run from Mathematica) for the computation of a Seifert matrix of a braid and the determinat of a braid as a polynomial in indeterminate exponents of the braid group generators. These programs were used for the computations in this paper and also in my previous papers. The texts of the programs are in the file  sm.mat. The user's guide is in the appendix of the paper. The examles considered in the user's guide are in the file  m8-exmpl.mat.