This is a link to a paper called Hyperbolic Celtic Knot Patterns.
It's pretty much what you're looking for.
The main class of problems I know of from knot theory is to determine whether two knots are equivalent up to manipulation, where the type of manipulation allowed is basically a formalisation of how one would normally try to untangle a knot in real life. I also remember hearing of a link with arithmetic sequences, although I don't remember details. According to Wikipedia, Gauss pioneered the mathematical study of knots and it eventually came to be considered as a subfield of topology. It could also show up in "recreational mathematics", that is, puzzles aimed at enthusiasts who aren't necessarily professional mathematicians. You might also be interested in studying Moebius strips and related topics.