The mathematical discipline of knot theory started from an idea by Lord Kelvin about what atoms might be, from Britannica:

The reason the study of knots fell under the aegis of topology had to do with the fact that the essential structure of a knot is only how it is connected, and that doesn't change when you smoothly stretch or distort the knot. The knots can be decomposed into prime knots, similar to how all integers can be decomposed into prime numbers, here are the first 15 prime knots:

The study of knots turned out to be a dead end for atomic theory but it became useful in biology and materials science. In biology, the knots structures have applications in studying protein folding, and in materials science, there are topological phases of matter that have novel properties, properties that are best understood by studying the order of the state changes of a system, since that order creates a different knot through spacetime. It's fascinating stuff, and is informing a direction of research in quantum computing that Microsoft Research is pursuing. See the topological qubit.

I am not going to do an intro to knot theory yet, but I will share these files:

I've been 3D printing knots and studying them in physical 3D space. I was inspired by @algekalipso's knot-meditation practice.

Back on 2024-01-09, I wrote the brief Phenomenal binding problem and topology where @algekalipso explains his ideas about the connections between consciousness and topology. His meditation practice involves meditating on the structure of the prime knots, but I don't personally have a meditation practice, however, I do pray, and it just so happens that my choice of image for my prayer corner is Our Lady, Undoer of Knots (print by Daniel Mitsui):

So I began using the prime knots as objects to focus my attention and ask Our Lady for help in untying them. What's mysterious about knot theory is that it only works in 3 dimensional space! 2 dimensional space doesn't allow crossings, notice in the prime knots image, how there is a gap, that gap represents a crossing in 3D that gets projected down as an intersection in 2D. There is a proof that all knots can be continuously undone in 4D. It's befitting of a spiritual being to be able to move in 4D space.

Finally, all by accident, I discovered my cat doesn't mind the 3₁ prime knot (Trefoil knot), but for some reason hates the 4₁ prime knot (Figure Eight knot):

Whether or not cat approval is a topological invariant of these knots is still an open question.