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In page-340 of Tristan Needham's Visual Complex Analysis, he introduces a theorem to find number of a loop. To motivate the theorem, he shows the situation of a point coming closer and closer to a segment of a curve:

enter image description here

For the right most picture, he writes a relation between winding numbers:

$$ v(K,r) = v(K,s) + v(L,s)$$

Or,

$$ v(K,r) = v(K,s) -1 $$

Or,

$$ v(K,r) +1 =v(K,s) \tag{1}$$

Explanation given:

enter image description here

Start from outside L, where you know the winding number is zero, move from region to region using crossing rule to add or subtract one at each crossing of L

The equation (1) and the idea behind its derivation allows us to relate winding numbers between the interior and exterior of the sets in which the curve partitions the plane. However, how can this be used to figure out the winding numbers easily?

Note:

$v(k,s) $ means the winding number of the loop $k$ around the point $s$

tryst with freedom
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2 Answers2

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Let's think of $L$ as a one way road. The equation states that when crossing the road, your winding number increases by one if traffic is coming from your left and decreases by one if traffic is coming from your right.

Now look at the example: Starting at the exterior region $D_4$ we have $v_4=0$. To get into $D_3$ we have to cross the road with traffic coming from our left, so the winding number increases to $v_3=1$. From $D_3$ to $D_2$ we have to cross $K$ with traffic coming from our right, so $v_2=v_3-1=1$. From $D_3$ to $D_1$ we cross $K$ with traffic coming from our left, so $v_1=v_3+1=2$.

Christoph
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  • For this, first we need to choose orientation of the curve. So, is the winding number not intrinsic to curve rather how it is oriented? – tryst with freedom Dec 18 '20 at 16:45
  • Secondly how did you conclude that in $D_4 $ the winding number is zero – tryst with freedom Dec 18 '20 at 16:45
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    Yes, the winding number is a property of an oriented curve. The sign is chosen such that a counter-clockwise circle around a point has winding number $+1$ for that point. For a point in $D_4$, there is a ray starting in that point that doesn't intersect $K$ at all, hence $K$ does not wind around the point. – Christoph Dec 18 '20 at 16:48
  • " for that point. For a point in D4, there is a ray starting in that point that doesn't intersect K at all, hence K does not wind around the point. " where did the ray thing come from? – tryst with freedom Dec 18 '20 at 16:59
  • What's your definition of winding numbers? – Christoph Dec 18 '20 at 16:59
  • You say how it increases as we move along roads, but the whole road-track is what we define winding number for , no? – tryst with freedom Dec 18 '20 at 16:59
  • The book says this "winding number v(L,0) of a closed loop L about origin O is the net number of revolution of the direction of z as it traces out L in a given sense" – tryst with freedom Dec 18 '20 at 17:00
  • We define winding numbers $v(K,p)$ of a curve $K$ relative to a point $p$. Here we are fixing $K$ and varying $p$. – Christoph Dec 18 '20 at 17:00
  • Yeah that part I understood, but the thing I don't getis about the winding number of the $D_4$ – tryst with freedom Dec 18 '20 at 17:01
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    Have a look at the animation in the Wikipedia article on Winding numbers — if your are standing outside of the curve and trace the curve with your eyes, you will never do a full revolution since there is a direction in which there are no points of the curve. That direction is the ray I referred to in the previous comment. – Christoph Dec 18 '20 at 17:03
  • OH I finally got it I think. So the curve partitions the plane into sets, for each set there is a winding number associated .. like each point in a set has the same winding number? @Christoph – tryst with freedom Dec 19 '20 at 12:53
  • That's right. Moving the point without crossing K won't change the winding number, hence it is constant on these regions! – Christoph Dec 19 '20 at 14:58
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Simple answer which I didn't pick up on when I read this originally: The result is super useful due to two facts.

  1. The curve is split the whole plane into some number of domains. In each of these domains, the winding number is constant.

  2. Using this rule, just knowing the winding number is one domain is enough to figure out winding number of any other

tryst with freedom
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