Serge Tabachnikov

Invariants of Legendrian and transverse knots in the standard contact space

1.1. The standard contact structure in 3-space, which arises from the identification of R3 with the manifold of l-jets of smooth real functions of one variable, naturally distinguishes two major classes of smooth immersed spatial curves: Legendrian curves, which are integral curves of the contact distribution, that is everywhere tangent to the distribution, and transverse curves, which are nowhere tangent to it. Closed embedded Legendrian and transverse curves are called Legendrian and transverse knots. Theories of Legendrian and transverse knots, which are clearly related to each other, are parallel to the classical knot theory in space. Legendrian and transverse knots have become very popular in contact geometry since the seminal work of Bennequin [3], published in 1983. For Legendrian knots one introduces two integer-valued Legendrian isotopy invariants. The first measures the rotation of an (oriented) knot with respect to the contact distribution; we call it the Maslov number. The second one, which we call the Bennequin number, is defined as the contact self-linking number of the knot. (For exact definitions of these and subsequent notions see Section 2.) Transverse knots have no Maslov numbers, but also have Bennequin numbers. The main achievement of Bennequin’s paper consists in two inequalities for these numbers (see Theorem 2.3 below), which imply, in particular, that the Bennequin number of a topologically unknotted Legendrian knot must be always negative. In turn, this gives rise to a construction of an exotic contact structure in R3 (or rather to a proof, that some previously known contact structures in R3 are not diffeomorphic to the standard one). Bennequin and Maslov numbers may be also used for distinguishing Legendrian or transverse isotopy classes of knots within a topological isotopy class. It is very easy to show that any topological knot is isotopic to (actually is Co approximated by) both Legendrian and transverse knots. It is equally easy to construct topologically isotopic Legendrian or transverse knots with different Bennequin and Maslov (in the Legendrian case) numbers. Since no other specifically Legendrian or transverse invariants of knots have been found so far, one may expect that topologically isotopic Legendrian knots with equal Bennequin and Maslov numbers are Legendrian isotopic, and similarly for transverse knots. The results of this article may be regarded as a confirmation of this conjecture.

Chapter 13 Rational billiards and flat structures

Publisher Summary The theory of mathematical billiards can be partitioned into three areas: convex billiards with smooth boundaries, billiards in polygons (and polyhedra), and dispersing and semi-dispersing billiards (similarly to differential geometry in which the cases of positive, zero, and negative curvature are significantly different). These areas differ by the types of results and the methods of study. This chapter illustrates the unfolding procedure in the simplest example of a rational polygon, the square. The chapter examines certain examples of Teichmtiller discs that arise in the so-called Veech billiards and their generalizations. Some results on ergodicity of vertical foliations of quadratic differentials are described in the chapter. The chapter presents the constructive proofs of other results on polygonal billiards and flat surfaces. In particular, it gives a new proof of the quadratic upper bound on the number of saddle connections.

Topological robotics: motion planning in projective spaces

In this paper, we study one of the most elementary problems of the topological robotics: rotation of a line, which is fixed by a revolving joint at a base point. One wants to bring the line from its initial position A to a final position B by a continuous motion in space. The ultimate goal is to construct a motion planning algorithm which will perform this task once the initial position A and the final position B are presented. This problem becomes hard when the dimension of the space is large. Any such motion planning algorithm must have instabilities, that is, the motion of the system will be discontinuous as a function of A and B. These instabilities are caused by topological reasons. A general approach to study instabilities of robot motion was suggested recently in [6, 7]. With any path-connected topological space X, one associates in [6, 7] a number TC(X), called the topological complexity of X. This number is of fundamental importance for the motion planning problem: TC(X) determines character of instabilities which have all motion planning algorithms in X. The motion planning problem of moving a line in R n+1 reduces to a topological problem of calculating the topological complexity of the real projective space TC(RP n ), which we tackle in this paper. We compute the number TC(RP n ) for all n ≤ 23 (see

Liouville–Arnold integrability of the pentagram map on closed polygons

The pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras, and integrable systems. Integrability of the pentagram map was conjectured by Schwartz and proved by the present authors for a larger space of twisted polygons. In this article, we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons. In the case of convex polygons in the real projective plane, this result implies the existence of a toric foliation on the moduli space. The leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic. Our proof is based on an invariant Poisson structure on the space of twisted polygons. We prove that the Hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants.

Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards

We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in R m+1 for m ? 3. For plane billiards (when m = 1) such bounds were obtained by Birkho5 in the 1920s. Our proof is based on topological methods of calculus of variations — equivariant Morse and Lusternik– Schnirelman theories. We compute the equivariant cohomology ring of the cyclic con guration space of

Billiards in Finsler and Minkowski geometries

Abstract We begin the study of billiard dynamics in Finsler geometry. We deduce the Finsler billiard reflection law from the “least action principle”, and extend the basic properties of Riemannian and Euclidean billiards to the Finsler and Minkowski settings, respectively. We prove that the Finsler billiard map is a symplectomorphism, and compute the mean free path of the Finsler billiard ball. For the planar Minkowski billiard we obtain the mirror equation, and extend the Mather’s non-existence of caustics result. We establish an orbit-to-orbit duality for Minkowski billiards.

Tire track geometry: Variations on a theme

We study closed smooth convex plane curves Λ enjoying the following property: a pair of pointsx, y can traverse Λ so that the distances betweenx andy along the curve and in the ambient plane do not change; such curves are calledbicycle curves. Motivation for this study comes from the problem how to determine the direction of the bicycle motion by the tire tracks of the bicycle wheels; bicycle curves arise in the (rare) situation when one cannot determine which way the bicycle went.We discuss existence and non-existence of bicycle curves, other than circles; in particular, we obtain restrictions on bicycle curves in terms of the ratio of the length of the arcxy to the perimeter, length of Λ, the number and location of their vertices, etc. We also study polygonal analogs of bicycle curves, convex equilateraln-gonsP whosek-diagonals all have equal lengths. For some values ofn andk we prove the rigidity result thatP is a regular polygon, and for some we construct flexible bicycle polygons.

More on Paperfolding

It is a common knowledge that folding a sheet of paper yields a straight line. We start our discussion of paperfolding with a mathematical explanation of this phenomenon. The model for a paper sheet is a piece of the plane; folding is an isometry of the part of the plane on one side of the fold to another, the fold being the curve of fixed points of this isometry (see Figure 1). The statement is that this curve is straight, that is, has zero curvature.

Contact Complete Integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical ℝ ⋉ ℝn−1 structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures.We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories.

The Poncelet Grid and Billiards in Ellipses

closure theorem (or Poncelet porism) is a classical result of projective geometry. Given nested ellipses y and I\ with y inside T, one plays the following game: starting at a point xonT, draw a tangent line to y until it intersects F at point y, repeat the con struction, starting with y, and so on. One obtains a polygonal curve inscribed in V and circumscribed about y. Suppose that this process is periodic: the nth point coincides with the initial one. Now start at a different point, say x\. The Poncelet closure theo rem states that the polygonal line again closes up after n steps (see Figure 1). We call these closed inscribed-circumscribed curves Poncelet polygons. Although the Poncelet theorem is almost two hundred years old,1 it continues to attract interest (see [2], [3], [4], [5], [6], [9], [12], [14], [15], [19], [23], [24] for a sample of references).