Figuring out winding number of a curve using crossing rule of a point

Question

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:

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:

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 $s$

Answer 1

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 $v_1=v_3+1=2$.

Answer 2

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