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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 $s_{+}D$:

enter image description here

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.

enter image description here

So what does the author mean by "$\cdots$ boundaries of the regions of one of the colors".

Can someone explain it to me, please?

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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:

Face types for alternating diagram

I am calling them type R and type L. Notice that around a crossing, type R and type L regions alternate:

Face types alternate around a crossing

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:

Smoothed type R region gives circle

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.

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  • $\begingroup$ @Miller: Are we assuming that the alternating link diagram is non-split diagram? I am asking this because if we take the disjoint union of trefoil and its mirror image, then we are not getting the desired result. $\endgroup$
    – eyp
    Aug 14, 2019 at 16:28
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    $\begingroup$ @eyp I was assuming a non-split diagram, but there is a way to deal with split diagrams: the disjoint union of two adequate links is adequate, so reduce to the non-split case. (I think your example is one Lickorish might not have anticipated in his proof of Proposition 5.3, so good work.) By the way, I realize I only hinted at the answer to your original question: the circles of $s_+D$ correspond to boundaries of the black (or white) regions of the original diagram. In your first picture of a trefoil, there are two black regions and two circles. In the second, three white and three circles. $\endgroup$ Aug 14, 2019 at 17:06
  • $\begingroup$ @Miller: I didn't understand this part " so reduce to the non-split case." Can you please explain it? I understand the disjoint union of two adequate links is adequate. $\endgroup$
    – eyp
    Aug 17, 2019 at 3:51
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    $\begingroup$ @eyp If you want to prove that every reduced alternating diagram is adequate, you can induct on the number of components in the diagram (not link components, but splittable pieces of the diagram). The base case is a non-split diagram, and the induction is when there is a split diagram. In this case, if a reduced alternating diagram is split, both sides of the splitting circle are reduced alternating diagrams, too. So, by induction we know they are adequate, and then so is their disjoint union. $\endgroup$ Aug 17, 2019 at 4:57
  • $\begingroup$ @Miller: Thank you, Miller. $\endgroup$
    – eyp
    Aug 17, 2019 at 6:22

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