ONPS Summer undergraduate research program in Mathematics, Chengdu

Extrema of Arnold's invariants of curves on surfaces

Vladimir Chernov

An (immersed) curve on a surface is a curve whose velocity vector is nowhere zero. A regular homotopy of an immersed curve is its smooth deformation in the class of immersed curves. Whitney's Theorem says that two immersed curves are regular homotopic if and only if the rotation numbers of their velocity vectors are equal.

During a generic regular homotopy a curve experiences the following three basic modifications (peresotroikas):

1. a passage through a triple point at which no two branches are tangent;

2. a passage through a direct tangency point at which the two branches are tangent of order one and have the same directions;

3. a passage through an inverse tangency point at which the two branches are tangent of order one and have opposite directions.

It is possible to prescribe a sign ±1 to the direction of each one of these perestroikas.

Arnol’d has axiomatically defined three basic invariants of planar curves:

1. St (Strangeness) that increases by one under a positive triple point perestroika and does not change under two other perestroikas;

2. J⁺ that increases by two under a positive direct tangency perestroika and does not change under other perestroikas;

3. J^- that decreases by two under a positive inverse tangency peretroika and does not change under other perestroikas.

He has also made conjectures about the range of values of the three invariants of planar curves and the curves on which the extremal value are attained.

The explicit formulas for the invariants were discovered and the conjectures were solved by Shumakovich and Viro. Different formulas through Gauss diagrams were later and independently discovered by Polyak.

The three Arnol’d invariants were generalized to curves on other surfaces in the works of Chernov and Inshakov. The generalized invariants are defined up to an additive constant for each regular homotopy class. So that the range of values of the invariants is not well defined, but the curves where the invariants attain extremal values are well defined.

It would be interesting to know what are the curves on surfaces (within a given regular homotopy class) on which the three invariants achieve the extremal values. In particular, if the regular homotopy class contains planar curves, is it true that the curves where the extremal values of the invariants are attained are the same as in the case of the planar curves?

Also for Strangeness it would be interesting to know if there are local extremal curves such that the value of St on a curve is bigger (respectively smaller) then the values of Strangeness obtained from the curve by all possible triple point moves.

The goal of this undergraduate research program is to obtain numerical data to formulate the conjectures on the shape of curves where the three basic invariants of curves on surfaces attain the extremal values.

Bibliography

V. I. Arnolʹd, Topological invariants of plane curves and caustics. Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. University Lecture Series, 5 American Mathematical Society, Providence, RI (1994) viii+60 pp. ISBN: 0-8218-0308-5

V. I. Arnolʹd, Plane curves, their invariants, perestroikas and classifications. With an appendix by F. Aicardi. Adv. Soviet Math., 21, Singularities and bifurcations, 33--91, Amer. Math. Soc., Providence, RI, (1994) also available as http://www.pdmi.ras.ru/~arnsem/Arnold/Arnold-PC93.pdf

A. V. Inshakov, Invariants of types j+, j− and st of smooth curves on two-dimensional manifolds. (Russian) Funktsional. Anal. i Prilozhen. 33 (1999), no. 3, 35--46, 96; translation in Funct. Anal. Appl. 33 (1999), no. 3, 189--198 (2000)

also available as http://link.springer.com/article/10.1007%2FBF02465203

M. Polyak, Invariants of curves and fronts via Gauss diagrams. Topology 37 (1998), no. 5, 989--1009. also available as http://www2.math.technion.ac.il/~polyak/publ/front.ps.gz

A. Shumakovich, Explicit formulas for the strangeness of plane curves. (Russian) Algebra i Analiz 7 (1995), no. 3, 165--199; translation in St. Petersburg Math. J. 7 (1996), no. 3, 445--472, also available as http://www.math.dartmouth.edu/~chernov/strangeness.pdf

V. Tchernov (Chernov), Arnold-type invariants of curves on surfaces. J. Knot Theory Ramifications 8 (1999), no. 1, 71--97, also available as http://arxiv.org/abs/math/9906125

O. Viro, Generic immersions of the circle to surfaces and the complex topology of real algebraic curves. Topology of real algebraic varieties and related topics, 231--252, Amer. Math. Soc. Transl. Ser. 2, 173, Amer. Math. Soc., Providence, RI, (1996) also available as http:/www.pdmi.ras.ru/~olegviro/gudk.ps