Are the following two links isomorphic?

Question

Please consider the following links with four components

My question is if such two links are isomorphic.

The corresponding Jones polynomials are respectively

It is observed that the ratio of the Jones polynomials is $q^{9⁄2}$. It is to say the only difference between the two Jones polynomials is a simple monomial.

According with such fact I am concluding that the two links are isomorphic. Do you agree?

The first link is as follows

Answer 1

In general, finding two links which have (almost in this case) same Jones polynomial does not imply they are the same link.

Using SnapPy, a program by Marc Culler and Nathan Dunfield, we can get a computer to answer some of these questions for us.

You can draw the links you drew and here they are, so you can check that I didn't make a mistake.

I called the first one $N$ and the second one $M$, which you can see at the top of the windows.

Using some of the built in operations, we can try to see if they are the same link.

Here is another image which shows us some of the computations SnapPy has done.

As you can see, we are getting different volumes and fundamental groups for $N$ and $M$. SnapPy has a "are these the same manifold" command, which I used at the bottom of the window, but this did not complete in the standard runtime. And the identify command works on $M$ and not on $N$.

Now, I have not proven these are actually different. It is possible SnapPy has just failed to recognize these are the same, but it is highly suspect. Out of the invariants I computed here, volume is the one I would probably trust the most, and since it is so far off, 7.3 to 14.6, I doubt they are the same link. You just happened to find two links with very similar Jones polynomials.

EDIT:

Having made the change you showed in your edit, here is the new link, just to make sure.

And here is the identification via SnapPy, which shows that $N$ is $L10a169$. So it is definitely not the link $M$, which is $L14n62566$.

Answer 2

I am making many computations of invariants for the two links using the Mathematica package KnotTheory. I am obtaining that the two links have different signatures. I am concluding that the two links are almost isomorphic. Moreover, the mirror images of the two links are isomorphic. Do you agree?

Answer 3

Using SnapPy with the first link as

and using the following code

M=Manifold()
M.volume()
M.fundamental_group()
M.identify()

I am obtaining the following output

14.655449507

Generators:
   a,b,c,d
Relators:
   adBAcaCDbcABdcACabDAcccACadBAcaCDbaCBdcACabDCC
   aCBcACadbcACaBAcaCDbAc
   adbcACaDAcaCdBAcaCDcAC

[L14n62566(0,0)(0,0)(0,0)(0,0)]

Note that the volume for the first link is 14.655449507 and the volume for the second link according with N. Owad is 14.65544951. The results are near identical and then I am concluding that the two links have the same volume.

For other side SnapPy identifies the first link as L14n62566 and according with N. Owad the second link is also L14n62566.

Then we conclude that the two links are the same. Do you agree?

Answer 4

Now I am using the PD codes for the two links. I am using SnapPy with the following code

In[1]: LS1=Link([[1,18,2,5],[5,2,6,3],[3,13,4,12],[12,4,11,1],[32,36,31,33],[33,31,34,30],
[23,34,24,35],[35,24,36,25],[26,10,25,11],[27,14,26,13],[6,28,7,27],
[18,32,17,19],[19,29,20,28],[7,20,8,21],[9,15,10,14],[17,30,16,29],[21,8,22,9],[22,16,23,15]])


In[2]: LS2=Link([[8,2,7,1],[26,3,25,2],[22,6,21,1],[8,18,9,17],[19,28,18,21],
[27,9,28,10],[3,16,4,15],[16,26,17,27],[14,25,15,24],[7,14,12,13],
         [13,23,20,22],[24,11,23,12],[6,20,5,19],[11,4,10,5]])

In[4]: LSI1=LS1.exterior()

In[5]: LSI2=LS2.exterior()

In[7]: LSI1.volume()
Out[7]: 14.655

In[8]: LSI2.volume()
Out[8]: 14.66

In[10]: LSI1.identify()
Out[10]: [L14n63195(0,0)(0,0)(0,0)(0,0)]

In[11]: LSI2.identify()
Out[11]: [L14n63195(0,0)(0,0)(0,0)(0,0)]

According with the Out[10] and Out[11] the two links are the same, namely L14n63195.

But at the same time SnapPy also produces L14n62566.

In conclusion SnapPy says that the two links are the same but with two different identifications L14n63195 and L14n62566.

When the computations are performing using the pictures of the two links the result is L14n62566.

Could you confirm my computations?

The following figures are showing that the two links are the same