User:Georg-Johann/sandbox

Let $$B(a, b)$$ be the image of the region $$a \leqslant \|w\| \leqslant b$$ of the Bryant-Kusner parametrization, and let $$\tau$$ be the positive, real solution of $$\tau^6+\sqrt{5}\tau^3-1 = 0$$. Then
 * $$\tau = \sqrt[3]{\frac{3-\sqrt5}2} = \varphi^{-2/3}\approx 0.72556$$

where $$\varphi$$ denotes the golden ratio, and the triple point is generated by the three $$w$$-values that satisfy $$\|w\|=\tau$$. Moreover:


 * $$B(0 < a < b < \tau)$$ is a circular, untwisted and un-knotted band.
 * $$B(0 = a < b < \tau)$$ is a cap.
 * $$B(\tau < a < b < 1)$$ is a band with six half-twists in the shape of a trefoil knot. As $$b$$ approaches 1, one edge of that band approaches itself.
 * $$B(\tau < a < b = 1)$$ is an un-knotted Möbius band with three half-twists.

All of the above shapes are smooth, have no self-intersections or other singularities, and have 3-fold rotational symmetry.