This is an exercise from the conway book. Here n(γ;a) is the winding number of γ around a. I have no idea how to find such a curve. To me this exercise seems to require a curve with arbitary winding numbers... Could anyone help me how to find such a curve?
On
We know that anything simple will fail - Conway tells us that winding number will be constant on connected components of the complement of $\gamma$. So we'll need to look for something that cuts $\mathbb{C}$ into infinitely many pieces.
The Hawaiian earring seems to be a solution to me - let $\gamma:[0,1]\to \mathbb{C}$ be the curve such that $\gamma([1-\frac{1}{2^{t}},1-\frac{1}{2^{t+1}}])$ is the circle of radius $\frac{1}{2^t}$, starting at the origin (and, of course, set $\gamma(1)=0$).
The image on the Wikipedia page is a good illustration: https://en.wikipedia.org/wiki/Hawaiian_earring
Indeed the earring cuts $\mathbb{C}$ into infinitely many pieces (The 'outside' of each circle inside the next-largest circle). To get a winding number of one, we pick a point inside the largest circle but outside the second-largest. To get a winding number of two, we pick a point inside that second-largest circle but inside the third-largest, and so on.
So heuristically this seems to work, we just need to make sure that we carefully check that this curve is both 1.) Continuous, and 2.) rectifiable. Rectifiability is fine - the lengths of the circles' circumferences shrink like $\frac{1}{2^t}$, so their sum of lengths converges. I'll leave you to check continuity, but it's not too bad (careful at $s=1$)
This takes care of $n\in \mathbb{N}$; to get negative winding numbers we should tack onto the end of $\gamma$ another Hawaiian earring going in the other direction, traversed clockwise. (The resulting image looks like a pair of glasses.)
Does the following picture resonate with you at all? Basically you've got a bounded collection of points $(a_i)_{i \in \Bbb Z}$, situated along a straight line, and you draw a curve so that it winds around each $a_i$ exactly $i$ times. Of course, you need to be careful about continuity near the accumulation points of the $a_i$, and you also nee to be careful with the rectifiability, but by keeping track of the diameter of the loops and making sure they decrease fast enough with $i$, those will not pose serious concerns.