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Gabai states that the Murasugi sum of two hopf bands yields a spanning surface of either the figure eight knot, the trefoil knot or a link of three components. Figure one shows two oppositely twisted hopf bands murasugi summed together to give the figure eight knot, I believe. If I Murasugi sum two hopf bands with twists in the same sense, using the summing disk in the same way as shown in figure one, then I get a spanning surface for the trefoil. Now if I rotate one of the hopf links in the diagram by 90 degrees and Murasugi sum in this way, regardless of whether the two links are twisted in like or opposite sense, I get the spanning surface of link of two components, not three. Is it a typo or have I not grasped the operation? Probably the later as the resulting surface seems to branch...

Gabai illustrates the Murasugi sum with "Figure 1":

Gabai figure 1

A sub question that might help clear this up is whether the disk in figure one is a 2-gon, (reasoning that the two edges that are an arc component of the link are what is being counted), or a 4-gon, (counting the two link edges and the two edges that are interior to the spanning surfaces). It seems to me that it must be a 4-gon as Gabai states that when the disk is a 2-gon the Murasugi sum is known as connected sum. What is depicted doesn't accord with my traditional understanding of a connected sum. However if summing as a connect sum were what was really meant in the case of the 2-gon, then this would avoid the problem of branching mentioned above, in the creation of a 2-component link. It would be at odds with condition one of definition 0.1: a surface is the murasugi sum of two sub surfaces if it is the union of these two surfaces identified over embedded disk D? (Or have I misconstrued the notation?) Connect sum seems to dispose of the disk...

I have tried to augment my understanding with Murasugi's discussion of S-surfaces given in 'On certain subgroup of the group of an alternating link'. I have also tried to think of where else it is possible to place the disk upon which we perform this generalised plumbing, however the 'translational symmetry' of the hopf band means that there is only one 'equivalence class' of locations we can place it.

How do I create a spanning surface for a 3-component link as the Murasugi sum of the surface of two hopf links?

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  • $\begingroup$ I edited the question and gave an answer, but this seems to me to be a math.SE question, because it's really far from research level. $\endgroup$
    – Daniel Moskovich
    Commented Feb 19, 2014 at 4:50
  • $\begingroup$ Noted. Will address such questions to appropriate site. Apologies and thanks. $\endgroup$
    – Lubtschenko
    Commented Feb 20, 2014 at 2:21

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The explanation of a Murasugi Sum in Chapter 4.2 of Kawauchi's book "A survey of knot theory" might clear up your doubts. "Figure 1", which you mention, indeed depicts a Murasugi Sum over a 4-gon.

To obtain a 3-component link, sum over a 2-gon, which has the effect of connect-summing two of the components (each twisted Hopf band has a 2-component link as its boundary) into 1.

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