Draw (2, 3) torus knot on the unwrapped torus surface

Question

I have constructed a (2, 3) torus knot using the parametric equation $$ \eqalignno{x =\,& \left(a + b \cos (qs)\right)\cos (ps)&{\hbox{(1)}}\cr y =\,& \left(a + b\cos (qs)\right)\sin(ps)&{\hbox{(2)}}\cr z =\,& b \sin (qs)&{\hbox{(3)}}} $$ where 0 ≤s≤ 2π. $a$ and $b$ are the major and minor radius of the torus. $p$ and $q$ are the integers to be varied to get different kinds of knots. The constructed knot on torus is shown here (2,3) knot side view and (2,3) knot top view. I wish to unwrap the virtual torus into a rectangle and plot the same knot on this rectangle. This is similar to the top picture in the link (3, 4) torus knot on the unwrapped torus surface, But I wish to do the same for the (2,3) torus knot in Matlab. I don't have an idea how to start with. I am looking for hints or reference links to understand. Any help or suggestion is appreciated.

Answer 1

Quite simply, note that the parameter space of the torus in which the knot is embedded is $$(\theta,\varphi) \in [0, 2\pi) \times [0, 2\pi),$$ and the torus itself is embedded in $\mathbb R^3$ as $$(x,y,z) = ((a+b \cos \varphi)\cos \theta, (a+b \cos \varphi) \sin \theta, b \sin \varphi).$$ So the knot corresponds to the choice $(\theta, \varphi) = (ps, qs)$. This immediately gives us the desired mapping, simply by plotting this as a parametric equation in $(\theta, \varphi) \in [0, 2\pi) \times [0, 2\pi)$; e.g., $$(\theta(s), \varphi(s)) = (ps, qs) \pmod {2\pi}, \quad s \in [0 , 2\pi).$$