The Multivariable Alexander Polynomial

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(For In[1] see Setup)

In[2]:= ?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.

In[3]:= MultivariableAlexander::about The multivariable Alexander program was written by Dan Carney at the University of Toronto in the summer of 2005.

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

In[5]:=	mva - (mva /. {t1->t2, t2->t3, t3->t4, t4->t1})
Out[5]=	-(t1 t2) + t2 t3 + t1 t4 - t3 t4

In[6]:=	mva - (mva /. {t1->t2, t2->t1})
Out[6]=	-(t1 t3) + t2 t3 + t1 t4 - t2 t4

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

In[7]:=	Select[AllLinks[], (MultivariableAlexander[#][t] == 0) &]
Out[7]=	{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]}