Plane real algebraic curves of odd degree with a deep nest

by S.Yu. Orevkov

Abstract:   In this paper we apply the Murasugi-Tristram inequality (as it was done in [1]) to study real algebraic curves of odd degree on  RP²   with a deep nest, i.e. curves which have a nest of the depth  k - 1  where   2k + 1   is the degree. We study also analogouse curves on real smooth ruled surfaces. For curves with a deep nest, the braid defined in [1] is uniquely determined by the arrangement of the curve with respect to the pencil of lines centered at a point inside the nest (the arrangement with respect to the fibers on the ruled surfaces).
    Moreover, if the degree is odd then the right hand side of Murasugi-Tristram inequality (the signature and the nullity of the braid) can be computed in some cases (and estimated in the other cases) inductively using the computations for iterated torus links due to Eisenbud and Neumann as the base of the induction and Conway's skein relation as the induction step.
    As an example of applications, we prove that some isotopy types are not realizable by M-curves of degree 9.
    In Appendix B, we give some generalization of the skein relation.

[1] S.Yu.Orevkov. Link theory and oval arrangements of real algebraic curves. Topology,   38(1999),  779-810
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