Dave Krebes
I enjoy studying knots. In my Ph.D. thesis (math), I show that a seamless unknotted loop (circle) of rope cannot be twisted, contorted or deformed in such a way as to intersect the interior of a sphere in the pair of arcs shown in the first picture (this is commonly known as a square knot). Equivalently: If a curve intersects the interior of a sphere in a square knot then it is genuinely knotted (ie. unalterably different from a circle). For example in the second picture the loop (follow it around-- it is indeed a single loop) is knotted: A long rubber band cannot be manipulated into this shape without breaking it and gluing the ends back together.
We call this a "topological" property of the square knot because it is a geometric property that is independent of lengths, angles, or rate of curvature (The size of the circle doesn't matter. In fact even an ellipse or a heart shape would do). The mathematical field of "algebraic topology", one of the great scientific achievements of the twentieth century, expresses many such properties in terms of boundaries. Thus to state the result in yet another way: Take a sphere (it could be egg-shaped) and a transparent, rubbery disc with a thick blue opaque circumference (boundary) and try to manoeuvre it (and the sphere too if you like) so that the picture inside the sphere (you can't see what the transparent interior of the disc is doing) is exactly as shown in the first picture, ie. a square knot. You will not succeed. However, even after you have tried many times and gained considerable experience with any of these variations of the problem, you must still organize your experience into a mathematical proof before you can conclude with certainty that you didn't stop just one twist and tug too soon! As a matter of fact, mathematicians were not satisfied until they had proven that the simplest of knots, the overhand knot (you can see two of these in the square knot above) is not in fact a slip-knot! This is something we all know from practical experience. Would you ever have thought of trying to prove such a statement?
As another exercise, try showing that if you replace each of the two strands of the square knot with two strands that run side-by-side like a pair of railroad tracks through the sphere, for four strands in total, then the situation is opposite to that described above: There is indeed such a disc, or a way to "tie" ("wrap" might be a better word; there are no free ends to tie with) the circle.
