- ga / 9 60 80 01 v 1 7 A ug 1 99 6 Bäcklund transformations and knots of constant torsion

The Bäcklund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vortex filament flow (also known as the Localized Induction Equation). In doing so, we obtain analytic constant-torsion representatives for a large number of knot types. Introduction Soliton equations have become a familiar presence in the differential geometry of curves and surfaces. The description of pseudo-spherical surfaces and their asymptotic lines in terms of the sine-Gordon equation dates back nearly a century. Of roughly the same date is the derivation (see [Ri]) of the Localized Induction Equation (LIE), which provides one of the richest examples of connection between curve geometry and integrability. The understanding of this connection has progressed in recent years along different directions. On the one hand, several fundamental properties of soliton equations have been given a geometrical realization; in the case of the LIE, its bihamiltonian structure and recursion operator, its hierarchy of constants of motion, and its relation to the nonlinear Schrödinger equation possess a natural geometric interpretation [L-P1]. On the other hand, some well-known classes of curves in differential geometry have been identified with solutions of integrable equations: for example, elastic curves and center-lines of elastic rods are among the solitons for the LIE, curves of constant torsion correspond to characteristics for the sine-Gordon equation, and planar and spherical curves are associated with solutions of the mKdV hierarchy ([G-P],[L-P2],[D-S]). A third direction of research ([C],[M-R]) concerns the topological properties of closed curves that arise as solutions to soliton equations. A major question is whether the presence of infinitely many symmetries and the associated sequence of integral invariants may be related to knot invariants; in fact, one hopes that the knot types of the periodic analogues of soliton solutions (the so-called multiphase solutions) can be described using methods of integrable systems, such as