The Multivariable Alexander Polynomial

2009-05-22T09:59:47Z

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{{Startup Note}}



{{HelpAndAbout|
n = 1 |
n1 = 2 |
in = <nowiki>MultivariableAlexander</nowiki> |
out= <nowiki>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.</nowiki> |
about= <nowiki>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.</nowiki>}}


{{Knot Image|L8a21|gif}}

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:



{{InOut|
n = 3 |
in = <nowiki>mva = MultivariableAlexander[Link[8, Alternating, 21]][t] /. {
t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3
}</nowiki> |
out= <nowiki>(-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])</nowiki>}}




{{InOut|
n = 4 |
in = <nowiki>mva - (mva /. {t1->t2, t2->t3, t3->t4, t4->t1})</nowiki> |
out= <nowiki>0</nowiki>}}




{{InOut|
n = 5 |
in = <nowiki>Simplify[mva - (mva /. {t1->t2, t2->t1})]</nowiki> |
out= <nowiki> (t1 - t2) (t3 - t4)
-----------------------------------
Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4]</nowiki>}}


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

====Links with Vanishing Multivariable Alexander Polynomial====

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



{{InOut|
n = 6 |
in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == 0) &]</nowiki> |
out= <nowiki>{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]}</nowiki>}}


{{Knot Image Quadruple|L9n27|gif|L10n32|gif|L10n36|gif|L10n107|gif}}
{{Knot Image Quadruple|L11n244|gif|L11n247|gif|L11n334|gif|L11n381|gif}}
{{Knot Image Triple|L11n396|gif|L11n404|gif|L11n406|gif}}

[[User:Drorbn|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:The Multivariable Alexander Polynomial|Talk Page]]).

====Detecting a Link Using the Multivariable Alexander Polynomial====

[[Image:Celtic-knot-basic-alternate.gif|thumb|right|200px|A mystery link]] On May 1, 2007 [[User:AnonMoos|AnonMoos]] asked [[User:Drorbn|Dror]] if he could identify the link in the figure on the right. So Dror typed:



{{InOut|
n = 7 |
in = <nowiki>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]</nowiki> |
out= <nowiki> 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] ))</nowiki>}}


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 <code>AllPossibilities</code> contain the multivariable Alexander polynomials of all those possibilities:



{{InOut|
n = 8 |
in = <nowiki>AllPossibilities = Union[Flatten[
{mva, -mva} /. {{}, {t[1] -> t[2], t[2] -> t[1]}}
]]</nowiki> |
out= <nowiki> 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] )}</nowiki>}}


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



{{InOut|
n = 9 |
in = <nowiki>Select[
AllLinks[],
MemberQ[AllPossibilities, MultivariableAlexander[#][t]] &
]</nowiki> |
out= <nowiki>{Link[11, Alternating, 289]}</nowiki>}}


And just to be sure,



{{InOut|
n = 10 |
in = <nowiki>{Jones[L][q], Jones[Link[11, Alternating, 289]][q]}</nowiki> |
out= <nowiki> -(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 }</nowiki>}}


Thus the mystery link is the mirror image of [[L11a289]].

Knot Atlas - User contributions [en]

The Multivariable Alexander Polynomial