|
|
 |
| |
|
|
|
Syllable Permutations and Hyperbolic Lorenz Knots |
|
|
|
PP: 2343-2348 |
|
|
|
Author(s) |
|
|
|
Paulo Gomes,
Nuno Franco,
Luís Silva,
|
|
|
|
Abstract |
|
|
| Lorenz knots are the knots corresponding to periodic orbits in the flow associated to the Lorenz system. This flow induces
an iterated one-dimensional first-return map whose orbits can be represented, using symbolic dynamics, by finite words. As a result of
Thurston’s geometrization theorem, all knots can be classified as either torus, satellite or hyperbolic knots. Birman andWilliams proved
that all torus knots are Lorenz knots which can be represented by a class of words with a precise form. We consider about 20000 words
corresponding to all non-trivial permutations of a sample of words associated to torus knots and, using the Topology and Geometry
software SnapPy, we compute their hyperbolic volume, concluding that it is significantly different from zero, meaning that all these
knots are hyperbolic. This leads us to conjecture that all knots in this family are hyperbolic. |
|
|
|
|
|
|
|
