I have a finite number of ordered points connected by a piecewise linear curve, which connects all the points in order and then the last with the first one. Can I check if there is a “knot” in this piecewise linear curve?
2026-03-27 14:02:09.1774620129
Check if there is a knot between piecewise linear line connecting finite number of points
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The answer is yes, but it can be difficult, depending on how tangled the curve is. I'm assuming that you want an automated method. Here are some things to try.
Find a planar projection with no intersections
For a curve that isn't too tangled, it may be possible to find a projection that has no intersections. This means the curve is an "unknot" (i.e. not a knot, contains no knots). If you can't find such a projection, your curve may or may not contain a knot.
Check whether the total curvature is less than $4\pi$
The Fáry–Milnor theorem states that smooth curves with small total curvature $(\le4\pi)$ must be unknotted. This generalizes to PL curves. If your PL curve has total curvature $\gt 4\pi$ it may or may not contain a knot.
Use knot theory techniques
Knot theory applies to both smooth curves and PL curves. One of the big preoccupations of knot theory is unknot recognition (and knot equivalence in general). Knots have associated polynomials, and the Jones Polynomial in particular can be used to distinguish most if not all knots from the unknot. A gentle introduction can be found in Conrad, The Knot Book. (can be temporarily "borrowed" from archive.org).
Use untangling techniques from computational geometry
As mentioned, even simple knots can be unrecognizable if they are sufficiently tangled. But there is software to address the problem. The most impressive I've seen is Yu, Schumacher, Crane, Repulsive Curves. Methods like this may help pre-process your PL curves so they are easier to use with the first three methods. In particular, if you have a gnarly unknot, Repulsive Curves should turn this into a nice circle or oval.