There's a classic puzzle that goes something like this:
You have two nails in a wall, and you want to hang a picture with a string (think of a necklace with a pendant) in such a way that if you remove either nail, the picture will fall to the floor.
One of the solutions that I know of is the so called Pochhammer contour. A more graspable picture can be found here. Here are a few questions as I am not very well versed in topology:
1) Is this solution unique, up to some homeomorphism? It's well known that if the nails are punctures, then the Pochhammer contour is homologous, but not homotopic to zero. So it seems like there's a topological notion lurking here which seems to be a necessary (but not sufficient?) condition for a curve to be a solution. For example, the winding number around each nail seems to necessarily be zero.
2) The solution curve $C$ allows for a very neat anlytic continuation of the beta function:
$$(1-e^{2\pi i a})(1-e^{2\pi i b})B(a,b)=\oint_C t^{a-1}(1-t)^{b-1}dt$$
I've never really seen a "motivated" derivation of above formula beyond just working out the integral and showing that it "works." I'm wondering if there's something connected here to picture frame puzzle, akin to removing one of the singularities? Or is it just a nice contour which has zero winding number around each singularity?