The Multivariable Alexander Polynomial
(For In[1] see Setup)
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![]() 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]:=
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mva = MultivariableAlexander[Link[8, Alternating, 21]][t] /. {
t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3
}
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Out[4]=
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-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
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In[5]:=
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mva - (mva /. {t1->t2, t2->t3, t3->t4, t4->t1})
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Out[5]=
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0
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In[6]:=
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mva - (mva /. {t1->t2, t2->t1})
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Out[6]=
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t1 t3 - t2 t3 - t1 t4 + t2 t4
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But notice the funny labelling of the components! At the moment there is no way to tell MultivariableAlexander
which variable is to be associated with what variable so MultivariableAlexander
chooses an arbitrary ordering of tha variables. Hence we had to rename t[3]
to be t4
and t[4]
to be t3
.
(To be precise, MultivariableAlexander
orders the components so that its output would be lexicographically minimal among all possible orderings. This way it is at least guaranteed that different presentations for the same link will yield the same output for MultivariableAlexander
.)
There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is . Here they are:
In[7]:=
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Select[AllLinks[], (MultivariableAlexander[#][t] == 0) &]
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Out[7]=
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{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]}
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![]() L9n27 |
![]() L10n32 |
![]() L10n36 |
![]() L10n107 |
![]() L11n244 |
![]() L11n247 |
![]() L11n334 |
![]() L11n381 |
![]() L11n396 |
![]() L11n404 |
![]() 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.