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Please consider the following brunnian link with four components

enter image description here

I am trying to identify such link using SnapPy but SnapPy is not able to do it.

My questions are :

  1. How many effective crosses such link have (apparently there are $14$ crosses but I think that the number of effective crosses is $12$)

  2. What link is such brunnian link?

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  • $\begingroup$ What is an "effective" cross? $\endgroup$
    – ypercubeᵀᴹ
    Commented Feb 27, 2019 at 13:46
  • $\begingroup$ @ypercubeᵀᴹ, is a cross that is not removable via a Reidemeister move of the type II. $\endgroup$
    – Juan Ospina
    Commented Feb 27, 2019 at 15:10

2 Answers 2

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I am not sure exactly what your first question is asking. You cannot remove any crossings from this diagram, as noted below by the fact that the link is "L14n..." and this means that the crossing number of your link is 14.

I am not sure why SnapPy failed to identify it for you, but when I tried, I got this, which answers your second question. Hope this helps.

In[46]: m.identify()
Out[46]: [L14n63195(0,0)(0,0)(0,0)(0,0)]

Here is my link in the plink editor. The Brunnian link in the question

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  • $\begingroup$ @ N. Owad, you are right. I was made a mistake. Many thanks for your answer. All the best. $\endgroup$
    – Juan Ospina
    Commented Feb 28, 2019 at 14:38
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Thanks to N. Owad, I am obtaining the folowing Jones polynomial for the link under consideration

enter image description here

The corresponding Khovanov-Poincaré polynomial is

enter image description here

enter image description here

Using SnapPy with the following code

 In[1]: LS=Link([[2, 24, 16, 1],[14, 4, 1, 5],[6, 15, 5, 16],[23, 6, 24, 7], [17, 14, 15, 13],[28, 22, 27, 23],[26, 13, 27, 12],[21, 25, 22, 28],[11, 26, 12, 25],[20, 8, 21, 7], [18, 10, 17, 11],[19, 20, 2, 3],[9, 18, 8, 19],[3, 4, 10, 9]])

    In[2]: LSI1=LS.exterior()

    In[3]:  LSI1.volume()
    Out[3]: 14.655

    In[4]: LSI1.identify()
    Out[4]: [L14n63195(0,0)(0,0)(0,0)(0,0)]

We identify the link as L14n63195.

It is worthwhile to note that the link under consideration is isomorphic to the following links

enter image description here

enter image description here

enter image description here

Please look at (Are the following two links isomorphic?)

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