A knot parametric equation typically refers to the equations used to describe a knot (in the context of knot theory) or the parametric representation of a curve that forms a knot. In most cases, we work with a three-dimensional curve that winds in a complicated way, and parametrize it using equations in terms of a single variable (typically denoted as tt).

For example, if you're looking for a parametric representation of a knot, one commonly used example is the trefoil knot.

Let me break down how you can generate a parametric equation for a knot:

1. Understanding the Knot

A knot can be represented as a closed curve in 3D space that is self-intersecting and non-trivial (i.e., it cannot be untangled without cutting). The most famous example is the trefoil knot.

2. Parametric Representation

A parametric equation for a knot uses a single parameter tt (which usually represents time or a parameter that runs along the curve) to express the coordinates x(t)x(t), y(t)y(t), and z(t)z(t) of the knot at each point in space.

For a trefoil knot, the parametric equations are:

x(t)=sin⁡(t)+2sin⁡(2t)x(t) = \sin(t) + 2 \sin(2t) y(t)=cos⁡(t)−2cos⁡(2t)y(t) = \cos(t) - 2 \cos(2t) z(t)=−sin⁡(3t)z(t) = -\sin(3t)

where tt is the parameter, typically varying from 0 to 2π2\pi.

3. Generating the Table

To create a table of values for the parametric equations, you would:

For example, here’s how you might generate some points for the trefoil knot:

tt x(t)x(t) y(t)y(t) z(t)z(t) 0 0 0 0 π6\frac{\pi}{6} ... ... ... π3\frac{\pi}{3} ... ... ... π2\frac{\pi}{2} ... ... ... ... ... ... ...

4. Steps to Create the Table

I think knotplot is fine. But I remember the max crossing number is 11or12 for this app.