Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

The Multivariable Alexander Polynomial

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(For In[1] see Setup)

In[1]:= ?MultivariableAlexander
MultivariableAlexander[L][t] returns the multivariable Alexander polynomial of a link L as a function of the variable t[1], t[2], ..., t[c], where c is the number of components of L. MultivariableAlexander[L, Program -> prog][t] uses the program prog to perform the computation. The currently available programs are "MVA1", written by Dan Carney in Toronto in the summer of 2005, and the faster "MVA2" (default), written by Jana Archibald in Toronto in 2008-9.
In[2]:= MultivariableAlexander::about
The multivariable Alexander program "MVA1" was written by Dan Carney at the University of Toronto in the summer of 2005; "MVA2" was written by Jana Archibald in Toronto in 2008-9.
L8a21.gif
L8a21

The link L8a21 is symmetric under cyclic permutations of its components but not under interchanging two adjacent components. It is amusing to see how this is reflected in its multivariable Alexander polynomial:

In[3]:= mva = MultivariableAlexander[Link[8, Alternating, 21]][t] /. { t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3 }
Out[3]= (-t1 - t2 + t1 t2 - t3 + 2 t1 t3 + t2 t3 - t1 t2 t3 - t4 + t1 t4 + 2 t2 t4 - t1 t2 t4 + t3 t4 - t1 t3 t4 - t2 t3 t4) / (Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4])
In[4]:= mva - (mva /. {t1->t2, t2->t3, t3->t4, t4->t1})
Out[4]= 0
In[5]:= Simplify[mva - (mva /. {t1->t2, t2->t1})]
Out[5]= (t1 - t2) (t3 - t4) ----------------------------------- Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4]

But notice the funny labelling of the components! The program MultivariableAlexander orders the variables in its output (typically denoted t[i]) in the same order as the order of the components of a link L as they appear within Skeleton[L]. Hence we had to rename t[3] to be t4 and t[4] to be t3.

Links with Vanishing Multivariable Alexander Polynomial

There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is 0. Here they are:

In[6]:= Select[AllLinks[], (MultivariableAlexander[#][t] == 0) &]
Out[6]= {Link[9, NonAlternating, 27], Link[10, NonAlternating, 32], Link[10, NonAlternating, 36], Link[10, NonAlternating, 107], Link[11, NonAlternating, 244], Link[11, NonAlternating, 247], Link[11, NonAlternating, 334], Link[11, NonAlternating, 381], Link[11, NonAlternating, 396], Link[11, NonAlternating, 404], Link[11, NonAlternating, 406]}
L9n27.gif
L9n27
L10n32.gif
L10n32
L10n36.gif
L10n36
L10n107.gif
L10n107
L11n244.gif
L11n244
L11n247.gif
L11n247
L11n334.gif
L11n334
L11n381.gif
L11n381
L11n396.gif
L11n396
L11n404.gif
L11n404
L11n406.gif
L11n406

Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots. (Though see the Talk Page).

Detecting a Link Using the Multivariable Alexander Polynomial

A mystery link
On May 1, 2007 AnonMoos asked Dror if he could identify the link in the figure on the right. So Dror typed:
In[7]:= mva = MultivariableAlexander[L = PD[ X[1, 16, 2, 17], X[3, 15, 4, 14], X[5, 8, 6, 9], X[7, 21, 8, 20], X[9, 22, 10, 13], X[11, 2, 12, 3], X[13, 18, 14, 19], X[15, 12, 16, 1], X[17, 11, 18, 10], X[19, 4, 20, 5], X[21, 7, 22, 6] ]][t]
Out[7]= 2 -(((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] + 2 t[1] t[2] - 2 2 2 2 2 2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] )) / 3/2 3/2 (t[1] t[2] ))

We don't know whether our mystery link appears in the link table as is, or as a mirror, or with its two components switched. Hence we let AllPossibilities contain the multivariable Alexander polynomials of all those possibilities:

In[8]:= AllPossibilities = Union[Flatten[ {mva, -mva} /. {{}, {t[1] -> t[2], t[2] -> t[1]}} ]]
Out[8]= 2 {-(((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] + 2 2 2 2 2 2 t[1] t[2] - 2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] 3/2 3/2 )) / (t[1] t[2] )), 2 ((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] + 2 t[1] t[2] - 2 2 2 2 2 2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] )) / 3/2 3/2 (t[1] t[2] )}

Finally, let us locate our link in the link table:

In[9]:= Select[ AllLinks[], MemberQ[AllPossibilities, MultivariableAlexander[#][t]] & ]
Out[9]= {Link[11, Alternating, 289]}

And just to be sure,

In[10]:= {Jones[L][q], Jones[Link[11, Alternating, 289]][q]}
Out[10]= -(17/2) 4 8 12 16 18 17 15 {q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q 10 3/2 5/2 ------- - 7 Sqrt[q] + 3 q - q , Sqrt[q] -(5/2) 3 7 3/2 5/2 -q + ---- - ------- + 10 Sqrt[q] - 15 q + 17 q - 3/2 Sqrt[q] q 7/2 9/2 11/2 13/2 15/2 17/2 18 q + 16 q - 12 q + 8 q - 4 q + q }

Thus the mystery link is the mirror image of L11a289.