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?