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 system↗MathWorldLorenzAttractor↗WikidataQ899844↗

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 $LR$ corners — the braid index of the link

Typed signatures

Type	Package	Applies
(string) -> expression<Braid>	braid	always

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 $L < R$
• The flow sends each rotation to the next, and the positive permutation braid of that permutation is the Lorenz braid
• Trip number $1$ means braid index $1$, which means the unknot — so $L^pR^q$ draws the UNKNOT, not $T(p,q)$
• When the permutation is a rotation $i \mapsto i+k$ on $n$ points the knot is $T(k, n-k)$; those words are the Christoffel words
• The shortest geodesic drawing a trefoil is $LLRLR$, of symbolic length 5
• A repeated word traverses one geodesic several times and has no well-defined point order, so it is refused

Examples

Basic #

the shortest knotted geodesic draws a trefoil

live ↗

In$\mathrm{AlexanderPolynomial}(\text{LLRLR})$Out$t^2-t+1$

Scope #

a rotation by 2 on 5 points — hence $T(2,3)$

live ↗

In$\mathrm{LorenzPermutation}(\text{LLRLR})$Out$\bigl\lbrack3, 4, 5, 1, 2\bigr\rbrack$

Properties #

braid index 2

live ↗

In$\mathrm{TripNumber}(\text{LLRLR})$Out$2$

one hump each way is unknotted, however long

live ↗

In$\mathrm{AlexanderPolynomial}(\text{LLLRRRR})$Out$1$

About

Examples	4
Crosswalk	WikipediaMathWorldWikidata

See also: Braid, AlexanderPolynomial, ModularClasses