I am reading the proof of the Proposition 5.3. of the chapter "The Jones Polynomial of an Alternating Link" from the book "Introduction to knot theory" by "Lickorish". I have a problem with understanding its proof. Before coming to the problem I will mention some background for the problem.
Suppose we are given a link diagram $D$ and at each crossing, we perform the following kind of smoothing and denoted as $s_{+}D$:
Now suppose we have an alternating link diagram with chessboard coloring.
Following line is written in the proof which I don't understand:
"The alternating condition implies that the components of $s_+ D$ are the boundaries of the regions of one of the colors (the black ones, say) with corners rounded off."
Following are the diagrams I have drawn after performing $s_+ D$ smoothing on trefoil and its mirror image. In both diagrams, each circle is the boundary of each color.
So what does the author mean by "$\cdots$ boundaries of the regions of one of the colors".
Can someone explain it to me, please?
Here's what this is meant to mean:
Take a look at a region of the alternating knot diagram, which is a disk. Due to it being an alternating diagram, regions come in only two types, depending on what the incident crossings look like:
I am calling them type R and type L. Notice that around a crossing, type R and type L regions alternate:
Thus, if we color all the type R regions black and the type L regions white, we have a checkerboard coloring. Now, if we do the $s_+$ smoothing, the black regions "have their corners smoothed" like so:
If we make sure the "outer" region is type L (there is always a diagram where this is the case --- it's simply by rotating the diagram on $S^2$, a.k.a. isotoping strands "through infinity"), then the upshot is the circles of $s_+D$ are not nested. Colored, it will be some number of black disks on a white plane.