ring knot lines as a locus from a mathematical function

[alana]

in looking for algebraic descriptions of ring knots,

it seems the 3Lx4B 'carrick mat' is a hypotrochoid roulette
https://mathcurve.com/courbes2d.gb/hypotrochoid/hypotrochoid.shtml
way-groovy animation diagrams

also the overhand knot 'silhouette' has a mathematical function that's named the 'knot curve'
 https://mathworld.wolfram.com/KnotCurve.html

so these aren't knot formulae, as the weave isn't specified,
but still, they look great and are a 2d match for knots.

any additional info is welcomed 

[KnotMe]

Knowing that the overhand is a trefoil knot, I was able to google up (trefoil knot parametric equation) this mathematician:
https://mathvis.academic.wlu.edu/tag/knots/#:~:text=The%20parametric%20equations%20of%20a,0.2%20cm%20around%20the%20curve.

from math stack exchange:
https://math.stackexchange.com/questions/577446/how-can-the-trefoil-knot-be-expressed-in-polar-coordinates

modelling in rhino:

more equations:
http://paulbourke.net/geometry/knots/

changing the search to turks head knot parametric equation we get a paper:
https://arxiv.org/pdf/math/0405151.pdf

IGKT's own Skip Pennock:
http://www.mi.sanu.ac.rs/vismath/pennock/index.html

http://symmetry-us.com/Journals/pennock1/index.html

A strangely handwritten paper that mentions THK many times:
https://personal.math.ubc.ca/~boyd/Apoly.symm.pdf

hmm, where did "billliard knots" come from?
https://mathcurve.com/courbes3d.gb/billardcylindrique/billardcylindrique.shtml

more THK math
https://www.facebook.com/notes/knot-tyers/text-for-how-to-tie-any-turks-head-knot/1788209474732709/

[alana]

many links, knotme...
i'm having a browse, thanks!

maths and calculus and the lot was a bit of a struggle for me in school, and beyond,
i liked the introductions to the various areas of maths such that i can recognise a bit of what's being said,

i love that there's a mathematical engine that drives these knots