[adeledeweylopez]

As a teen, I was obsessed with knot theory, and Conway's knot notation always felt like magic to me.

The notation works by counting the number of twists in a segment, and then looking for another twist directly connected to the previous twist, and so on.

This gives you a sequence of integers, one counting the number (and direction) of twists.

If the entire knot is made of twists connected this way, then the continued fraction you get from the sequence of numbers is a knot invariant!

https://www.maths.ed.ac.uk/~v1ranick/papers/conway.pdf

[whatshisface]

Can two different knots have the same integer sequence?

[kmill]

No, they can't, since they are just representations of the knot diagram.

As for uniqueness, even "rational knots," the ones from the simplest of the fundamental polyhedra, have multiple fractions representing them.

[sdenton4]

I'm knot familiar with this one in particular, but a 'unique' knot invariant would be a very big deal. (It would then be a knot identifier...)

[saagarjha]

;)