its a mathematical knot, which is essentially definable as a loop of "string" that has crossings that cannot be removed (ie the loop returned to a normal doughnut shape/torus) via any sort of transformation that doesn't involve cutting it.
the trefoil in the article is an example of such a thing.
Most knots we encounter in every day life are not a knot in a mathematical sense, because they can be untied, or don't form a loop.
I'm not familiar with the subject, so I don't quite get that last sentence. It seems to me like most knots I encounter in everyday life is similar to this one:
http://www.nationbydesign.com/simple-knot.jpg
This, to me, seems like a knot that cannot be removed via any sort of transformation that doesn't involve cutting it, assuming that the piece of string has no ends and continues forever.
Is my understanding of a knot in the mathematical sense wrong? Or of the involved transformations? If so, can you elaborate?
Your understanding is fairly correct, but it is mathematically easier/nicer to tie the ends together rather than let them run to infinity. That guarantees that you describe exactly how the ends would 'continue forever', and fits better with topology.
Take X = all continuous well behaved 1-1 mapping f from [0,1] to R^3, with f(0)=f(1).
A knot is an equivalence class of such functions, equivalence defined as given two functions f and g, you can "morph" one into other continuously - i.e. if you can find a parametrized well-behaved (i.e. continuous, differentiable etc.) function h(x;t) s.t. h(x;0)=f(x) and h(x;1)=g(x) and h(x;t_ is in X for all t.
It's pretty much the definition you'd come up with too.
This summarizes my entire experience with higher mathematics (and in fact academia in general).
To be fair, this is mostly because in order to have an in-depth conversation about a topic you first need to make sure everybody involved shares the same definitions. And in this case it's also because naming things is hard and "regular" knots are not very interesting in mathematics.
