enumeratio

@enumeratio/braid ​

Braid groups for compute-engine: the Artin presentation, positive braids and their closures, the Burau representation, Alexander polynomials, torus knots, and the Lorenz braids of modular geodesics. A braid word is a finite string of generator indices — a complete combinatorial name for the knot its closure ties. Start with knots and braids for the group and the closure construction, then torus knots for the simplest infinite family, and the Lorenz flow for the strange-attractor connection.

Declaring ​

ts
import { ComputeEngine } from "@cortex-js/compute-engine";
import { declareBraid } from "@enumeratio/braid";

const ce = new ComputeEngine();
declareBraid(ce);

Heads ​

Usage ​

JonesPolynomial(Braid(2, [1, 1, 1]))   // the right-handed trefoil's polynomial
AlexanderPolynomial(Braid(2, [1, 1, 1]))   // t^2 - t + 1 — same trefoil
AlexanderPolynomial("LLRLR")   // t^2 - t + 1 — the shortest knotted modular geodesic
TripNumber("LLRLR")   // 2 — its braid index

A modular geodesic word (L/R) and a braid closure can name the same knot; the Lorenz guide is where that correspondence comes from — see also @enumeratio/modular, whose S/T words feed the Lorenz braid construction.

Next ​

src/laurent.ts carries the Laurent-polynomial arithmetic the Alexander and Jones computations run on; src/curve.ts is the parametric-curve sampling behind the guides' 3D plots.

Pages ​