Symmetric knots and billiard knots

Abstract

Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but mathematically complicated, topological objects are Knots. We present in this papers several examples, both old and new, of regularity of algebraic invariants of knots. Our main invariants are the Jones polynomial (1984) and its generalizations. In the first section, we discuss the concept of a symmetric knot, and give one important example -- a torus knot. In the second section, we give review of the Jones type invariants. In the third section, we gently and precisely develop the periodicity criteria from the Kauffman bracket (ingenious version of the Jones polynomial). In the fourth section, we extend the criteria to skein (Homflypt) and Kauffman polynomials. In the fifth section we describe r^q periodicity criteria using Vassiliev-Gusarov invariants. We also show how the skein method may be used for r^q periodicity criteria for the classical (1928) Alexander polynomial. In the sixth section, we introduce the notion of Lissajous and billiard knots and show how symmetry principles can be applied to these geometric knots. Finally, in the seventh section, we show how symmetry can be used to gain nontrivial information about knots in other 3-manifolds, and how symmetry of 3-manifolds is reflected in manifold invariants. In particular, we find the formula for a torus knot in a solid torus and we show that the third eigenvector (e_3) of the Kauffman bracket skein module of the solid torus (with respect to the meridian Dehn twist) can be realized by a link in a solid torus.