Lissajous-toric knot

In knot theory, a Lissajous-toric knot is a knot defined by parametric equations of the form:

$$x(t)=(2+\sin qt)\cos Nt, \qquad y(t)=(2+\sin qt)\sin Nt, \qquad z(t)=\cos p(t+\phi),$$

where $$N$$, $$p$$, and $$q$$ are integers, the phase shift $$\phi$$ is a real number and the parameter $$t$$ varies between 0 and $$2\pi$$.

For $$p=q$$ the knot is a torus knot.

Braid and billiard knot definitions
In braid form these knots can be defined in a square solid torus (i.e. the cube $$[-1,1]^3$$ with identified top and bottom) as


 * $$x(t)=\sin 2\pi qt, \qquad y(t)=\cos 2\pi p(t+\phi), \qquad z(t)=2(N t - \lfloor N t\rfloor )-1, \qquad t \in [0,1]$$.

The projection of this Lissajous-toric knot onto the x-y-plane is a Lissajous curve.

Replacing the sine and cosine functions in the parametrization by a triangle wave transforms a Lissajous-toric knot isotopically into a billiard curve inside the solid torus. Because of this property Lissajous-toric knots are also called billiard knots in a solid torus.

Lissajous-toric knots were first studied as billiard knots and they share many properties with billiard knots in a cylinder. They also occur in the analysis of singularities of minimal surfaces with branch points and in the study of the Three-body problem.

The knots in the subfamily with $$p = q \cdot l$$, with an integer $$l \ge 1$$, are known as ′Lemniscate knots′. Lemniscate knots have period $$q$$ and are fibred. The knot shown on the right is of this type (with $$l=5$$).

Properties
Lissajous-toric knots are denoted by $$K(N,q,p,\phi)$$. To ensure that the knot is traversed only once in the parametrization the conditions $$\gcd(N,q)=\gcd(N,p)=1$$ are needed. In addition, singular values for the phase, leading to self-intersections, have to be excluded.

The isotopy class of Lissajous-toric knots surprisingly does not depend on the phase $$\phi$$ (up to mirroring). If the distinction between a knot and its mirror image is not important, the notation $$K(N,q,p)$$ can be used.

The properties of Lissajous-toric knots depend on whether $$p$$ and $$q$$ are coprime or $$d=\gcd(p,q)>1$$. The main properties are:
 * Interchanging $$p$$ and $$q$$:
 * $$K(N,q,p)=K(N,p,q)$$ (up to mirroring).


 * Ribbon property:
 * If $$p$$ and $$q$$ are coprime, $$K(N,q,p)$$ is a symmetric union and therefore a ribbon knot.


 * Periodicity:
 * If $$d=\gcd(p,q)>1$$, the Lissajous-toric knot has period $$d$$ and the factor knot is a ribbon knot.


 * Strongly-plus-amphicheirality:
 * If $$p$$ and $$q$$ have different parity, then $$K(N,q,p)$$ is strongly-plus-amphicheiral.


 * Period 2:
 * If $$p$$ and $$q$$ are both odd, then $$K(N,q,p)$$ has period 2 (for even $$N$$) or is freely 2-periodic (for odd $$N$$).

Example
The knot T(3,8,7), shown in the graphics, is a symmetric union and a ribbon knot (in fact, it is the composite knot $$5_1 \sharp -5_1$$). It is strongly-plus-amphicheiral: a rotation by $$\pi$$ maps the knot to its mirror image, keeping its orientation. An additional horizontal symmetry occurs as a combination of the vertical symmetry and the rotation (′double palindromicity′ in Kin/Nakamura/Ogawa).

′Classification′ of billiard rooms
In the following table a systematic overview of the possibilities to build billiard rooms from the interval and the circle (interval with identified boundaries) is given:

In the case of Lissajous knots reflections at the boundaries occur in all of the three cube's dimensions. In the second case reflections occur in two dimensions and we have a uniform movement in the third dimension. The third case is nearly equal to the usual movement on a torus, with an additional triangle wave movement in the first dimension.