The Multivariable Alexander Polynomial: Difference between revisions

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m (Reverted edits by LetodArlet (Talk); changed back to last version by Drorbn)
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n1 = 2 |
n1 = 2 |
in = <nowiki>MultivariableAlexander</nowiki> |
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.</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
⚫
about= <nowiki>The multivariable Alexander program was written by Dan Carney at the University of Toronto in the summer of 2005.</nowiki>}}
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>}}
<!--END-->
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t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3
t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3
}</nowiki> |
}</nowiki> |
out= <nowiki>-t1 - t2 + t1 t2 - t3 + 2 t1 t3 + t2 t3 - t1 t2 t3 - t4 + t1 t4 +
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</nowiki>}}
2 t2 t4 - t1 t2 t4 + t3 t4 - t1 t3 t4 - t2 t3 t4) /
(Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4])</nowiki>}}
<!--END-->
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n = 5 |
n = 5 |
in = <nowiki>mva - (mva /. {t1->t2, t2->t1})</nowiki> |
in = <nowiki>mva - (mva /. {t1->t2, t2->t1})</nowiki> |
out= <nowiki>t1 t3 - t2 t3 - t1 t4 + t2 t4</nowiki>}}
out= <nowiki>-((-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) /
(Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4])) +
(-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>}}
<!--END-->
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Line 98: Line 117:
X[19, 4, 20, 5], X[21, 7, 22, 6]
X[19, 4, 20, 5], X[21, 7, 22, 6]
]][t]</nowiki> |
]][t]</nowiki> |
out= <nowiki> 2 3 2
out= <nowiki> 2
-1 + 3 t[1] - 3 t[1] + t[1] + 3 t[2] - 7 t[1] t[2] + 7 t[1] t[2] -
-(((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] + 2 t[1] t[2] -
3 2 2 2 2
2 2 2 2 2
3 t[1] t[2] - 3 t[2] + 7 t[1] t[2] - 7 t[1] t[2] +
2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] )) /
3 2 3 3 2 3 3 3
3/2 3/2
3 t[1] t[2] + t[2] - 3 t[1] t[2] + 3 t[1] t[2] - t[1] t[2]</nowiki>}}
(t[1] t[2] ))</nowiki>}}
<!--END-->
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Line 113: Line 132:
n = 8 |
n = 8 |
in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == mva) &]</nowiki> |
in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == mva) &]</nowiki> |
out= <nowiki>{Link[11, Alternating, 289]}</nowiki>}}
out= <nowiki>{}</nowiki>}}
<!--END-->
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Revision as of 10:11, 21 April 2009


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

(For In[1] see Setup)

In[1]:= ?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. 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.
In[2]:= MultivariableAlexander::about
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.
L8a21.gif
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]:= mva - (mva /. {t1->t2, t2->t1})
Out[5]= -((-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) / (Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4])) + (-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])

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.)

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.gif
L9n27 		L10n32.gif
L10n32 		L10n36.gif
L10n36 		L10n107.gif
L10n107
L11n244.gif
L11n244 		L11n247.gif
L11n247 		L11n334.gif
L11n334 		L11n381.gif
L11n381
L11n396.gif
L11n396 		L11n404.gif
L11n404 		L11n406.gif
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

A mystery link

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] ))
In[8]:= Select[AllLinks[], (MultivariableAlexander[#][t] == mva) &]
Out[8]= {}

And just to be sure,

In[9]:= {Jones[L][q], Jones[Link[11, Alternating, 289]][q]}
Out[9]= -(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.

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