We've been discussing winding numbers in my complex course, and also Alexander polynomials and other invariants on knots in my Alg. Top. course, and the question came to me about the possibility of writing down the equation of a curve whose interior takes on every integer value as the winding number.
I believe Stein phrases it like: Given any closed rectifiable curve, for any integer $x$ there is a point $a$ interior to the curve with winding number $x$.
I picture it as an infinitely thick figure "8", because it should loop around itself infinitely many times in both directions, but I'm not actually sure that's possible. Does anybody have any ideas? I've researched some basic complex texts, like Stein or Alfors, but they only mention that such a curve exists, they don't produce one!
An explicit parametrization:
$$ \gamma(t) = \cases{\exp(1/t) (1 - \exp(-i/t^2)) & $-1/\sqrt{2\pi} \le t < 0$\cr 0 & $t = 0$\cr \exp(-1/t) (-1 + \exp(i/t^2)) & $0 < t \le 1/\sqrt{2\pi}$\cr}$$