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Knot Theory

A knot is an embedding of S1 in S3 (a 1-sphere in a 3-sphere, or a circle in three space plus the point at infinity), and a link is a disjoint collection of knots.
We generally think of a link as being a set of interlocking rings, such as a chain, but here we include unlinked sets, and the rings comprising them can be knotted. A periodic orbit of a continuous time dynamical system with two degrees of freedom is a knot since it is a closed loop embedded in three space. A subset of the set of periodic orbits of a system forms a link--in other words, the periodic orbits may be tangled up with each other in complex and interesting ways.

Chaotic dynamical systems are especially interesting from a knot theoretic point of view, as they have an infnite set of unstable periodic orbits that may be tangled in a way that includes every possible type of knot.

There are many ways to characterize knots and links that may be used to characterize the orbits of dynamical systems. Among these are the polynomial invariants:

  • Alexander Polynomial
  • Conway Polynomial
  • Jones Polinomial
  • HOMFLYPT Polynomial

    Suppose we have chaotic time series data from a black box and we want to determine the equations of the underlying dynamics in the box. We may reconstruct the phase space of the experimental system from its time series by the method of time delay embedding and extract the periodic orbits. With a few periodic orbits in hand, we may be able to characterize the dynamics of the system from the polynomial invariants of the knots and links that are the periodic orbits.

    My research in this area is an attempt to characterize the twisting dynamics of the manifolds of the periodic orbits of chaotic systems using some of the tools from the theory of knots, links, braids and templates. I would like to be able to translate a set of Poincare sections from a chaotic system and from that determine the knot templates on which the orbits evolve.
    I have included knot links for your amazement and edification.

  • Atlas of Oriented Knots and Links
  • Tie me up, tie me down
  • Knot Plot Site
  • Knots on the web
  • The knot square
  • knot glossary
  • Borromean Rings.
  • knotting index
  • knot index
  • alternating knots
  • Celtic knots
  • Hyperbolic knot
  • Intro to knot theory
  • knots
  • Knot Theory
  • Forensic knot analysis