The Multivariable Alexander Polynomial
From Knot Atlas
(For In[1] see Setup)
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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
}
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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])
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In[4]:=
| mva - (mva /. {t1->t2, t2->t3, t3->t4, t4->t1})
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Out[4]=
| 0
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In[5]:=
| Simplify[mva - (mva /. {t1->t2, t2->t1})]
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Out[5]=
| (t1 - t2) (t3 - t4)
-----------------------------------
Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4]
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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.
[edit] 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) &]
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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]}
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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).
[edit] Detecting a Link Using the Multivariable Alexander Polynomial
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]
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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] ))
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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]}}
]]
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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] )}
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Finally, let us locate our link in the link table:
In[9]:=
| Select[
AllLinks[],
MemberQ[AllPossibilities, MultivariableAlexander[#][t]] &
]
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Out[9]=
| {Link[11, Alternating, 289]}
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And just to be sure,
In[10]:=
| {Jones[L][q], Jones[Link[11, Alternating, 289]][q]}
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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 }
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Thus the mystery link is the mirror image of L11a289.
