1999/01/28 by Thomas A. Ivey, Thomas Ivey, Ivey, Thomas A. +2 · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Geometric and Algebraic Topology #Robotic Mechanisms and Dynamics #math.DG
paper · pdf · doi:10.48550/arxiv.math/9901131
17 pages, LaTeX, epsfig; to appear in Proc. London Math. Soc
arxiv created 1999/01/28 · arxiv updated 2009/11/30
The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the m-fold covered circle to the n-fold covered circle for any m,n relatively prime. Each family contains exactly one self-intersecting curve, one elastic curve, and one closed curve of constant torsion. Two torus knot types are represented in each family, and all torus knots are represented by elastic rod centerlines.