The Multivariable Alexander Polynomial
(For In[1] see Setup)


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 . 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 
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. (Though see the Talk Page).
Detecting a Link Using the Multivariable Alexander Polynomial
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.