enumeratio

Appearance

LorenzBraid ​

The braid a closed geodesic of the modular flow draws. By Ghys's theorem the modular knots are exactly the periodic orbits of the Lorenz attractor, and those have a purely combinatorial positive braid.

WikipediaLorenz systemMathWorldLorenzAttractorWikidataQ899844
LorenzBraid(word)the positive permutation braid of the orbit
LorenzPermutation(word)how the flow permutes the orbit's branch-line points
TripNumber(word)the count of corners — the braid index of the link

Domain: Braids and knots

Details
  • The word's cyclic rotations name the orbit's points; both branches of the template preserve orientation, so they are ordered lexicographically with
  • The flow sends each rotation to the next, and the positive permutation braid of that permutation is the Lorenz braid
  • Trip number means braid index , which means the unknot — so draws the UNKNOT, not
  • When the permutation is a rotation on points the knot is ; those words are the Christoffel words
  • The shortest geodesic drawing a trefoil is , of symbolic length 5
  • A repeated word traverses one geodesic several times and has no well-defined point order, so it is refused

Examples

See also: Braid, AlexanderPolynomial, ModularClasses