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
(available on this site)