|WikiProject Mathematics||(Rated B-class, High-importance)|
I don't think this article should be called a stub anymore. Though there's certainly more to be said about winding numbers, it is mostly in the context of how it's used with other techniques (residue theorem, etc.). The rest of the info should be put in those articles, not in this one.
The most i think this needs is a list of areas where the winding number is used.
Surely the circle is not homeomorphic to the plane minus one point, since the circle is compact and the plane minus a point is not. Should this say that they are homotopy equivalent? Martin Orr 13:33, 20 May 2006 (UTC)
What was meant by this? It clearly isn't the case that 2 curves with homeomorphic graphs have the same winding number around a fixed point. Is it trying to say that if γ is a closed path and w a point in C and f a homeomorphism of the plane, then thge winding number of f compose γ around w is I(γ,w)? Anyway I shall take it out for now.A Geek Tragedy 18:36, 6 February 2007 (UTC)
The "intuitve description" can be made rigorous and is then defined for ALL curves (not just closed or rectifiable). It's done that way in Beardons book on complex analysis and the argument principal. My preference would be to give that as the definition and state that the integral version is equvialent for closed, rectifiable curves (althouigh I think I only have a reference that it is equivalent in the case where the curve is pointwise continuously differentiable). A Geek Tragedy 18:36, 6 February 2007 (UTC)
A latest application and a question on the name
Recently a new application of the winding number was reported in the field of computed tomography,(Y Wei et al, Physical Review Letter 95, Paper no. 258102, 2005). In this paper, the winding number is further defined for some open curves with two asymptotes of the opposite directions (e.g. U shaped line, Fig 7a) or the same directions( straigt line, Fig 7b), and the winding number can be an integer plus a half. However,the winding number was called the turn number in this paper. Can the winding number be called the turn number or not? 220.127.116.11 08:33, 20 October 2007 (UTC)
minor things to be corrected
Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin index. - misleading? Is it not true for maps from n-sphere to n-sphere for any n? (the 'degree of a continuous map' article says that). Also, there is a reference to a picture on the right (illustrating the turning number), but no picture. 18.104.22.168 (talk) 03:59, 20 February 2008 (UTC)
- Re maps from n-sphere to n-sphere, yes, they are all also classified by a single integer. See Homotopy groups of spheres. —David Eppstein (talk) 04:02, 20 February 2008 (UTC)