design a mathematical knot in blender

Question

i'm a complete beginner to blender, i wanted to design various mathematical knots (e.g trefoil knot). What would be the best way of going about this, I've seen people design them freehand, but also saw much more precise results using formulae, any help/tips much appreciated.

Answer 1

There's a bunch of mathematical knot presets in the Torus Knot Plus object

Answer 2

One starting point might be

"Open Blender and go to Preferences then the Add-ons tab. Click Add Curve then Extra Objects to enable the script."

Although there are many options. You could start a new curve object and choose knots, and torus knot plus to experiment with:

Although not as quick a method, but producing results you can customize, the formula for the trefoil object can be plugged into animation and geometry nodes, taking the set up mentioned in this tutorial, just adapted for Geometry Nodes:

https://www.blenderdiplom.com/en/tutorials/simulation/671-trefoil-knot-animation-nodes-math-fun-pt-1.html

Answer 3

As others have mentioned, this is probably best addressed with the Extra Objects addon.

Since you specified you want to "design" these knots, however, there are ways to have more control. For instance, you can use the more general Shift + A Mesh → Math Function → X, Y, Z Function Surface strategy. An example for a trefoil know could look like this:

1. Define a trefoil knot surface using its implicit equations over a range of u values.

2. Delete all duplicate vertices with M → by distance

3. Give it some thickness using a skin + subsurf modifier, e.g.