The Multivariable Alexander Polynomial: Difference between revisions
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[http://wedner.info/content/view/tastiera-touchpad.htm tastiera touchpad] [http://soler.net.cn/data/bye-bye.htm bye bye baby baby goodbye] [http://funkall.us/images/small/portatile-registratori/ portatile registratori] [http://joperan.org/data/asian-fuck/ asian fuck] [http://svike.info/view/new/wu-ming.htm wu ming] [http://soler.net.cn/data/buste-imbottite.htm buste imbottite c] [http://polex.com.cn/resources/articles/elenco-abbonati.htm elenco abbonati cellulari] [http://sneck.info/data/bohr/ bohr] [http://veggis.org.cn/resources/articles/xfx-geforce.htm xfx geforce pci] [http://budget7i.info/lib/inglese-corso/ inglese corso intensivo] {{Manual TOC Sidebar}} |
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{{Manual TOC Sidebar}} |
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{{Startup Note}} |
{{Startup Note}} |
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| Line 26: | Line 26: | ||
t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3 |
t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3 |
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}</nowiki> | |
}</nowiki> | |
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out= <nowiki>-t1 - t2 |
out= <nowiki>-t1 - t2 t1 t2 - t3 2 t1 t3 t2 t3 - t1 t2 t3 - t4 t1 t4 |
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2 t2 t4 - t1 t2 t4 |
2 t2 t4 - t1 t2 t4 t3 t4 - t1 t3 t4 - t2 t3 t4</nowiki>}} |
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<!--END--> |
<!--END--> |
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| Line 44: | Line 44: | ||
n = 5 | |
n = 5 | |
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in = <nowiki>mva - (mva /. {t1->t2, t2->t1})</nowiki> | |
in = <nowiki>mva - (mva /. {t1->t2, t2->t1})</nowiki> | |
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out= <nowiki>t1 t3 - t2 t3 - t1 t4 |
out= <nowiki>t1 t3 - t2 t3 - t1 t4 t2 t4</nowiki>}} |
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<!--END--> |
<!--END--> |
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| Line 55: | Line 55: | ||
There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is <math>0</math>. Here they are: |
There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is <math>0</math>. Here they are: |
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<!--$$Select[AllLinks[], (MultivariableAlexander[#][t] == 0) |
<!--$$Select[AllLinks[], (MultivariableAlexander[#][t] == 0) |
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<!--Robot Land, no human edits to "END"--> |
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{{InOut| |
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n = 6 | |
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in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == 0) &]</nowiki> | |
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out= <nowiki>{Link[9, NonAlternating, 27], Link[10, NonAlternating, 32], |
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Link[10, NonAlternating, 36], Link[10, NonAlternating, 107], |
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Link[11, NonAlternating, 244], Link[11, NonAlternating, 247], |
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Link[11, NonAlternating, 334], Link[11, NonAlternating, 381], |
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Link[11, NonAlternating, 396], Link[11, NonAlternating, 404], |
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Link[11, NonAlternating, 406]}</nowiki>}} |
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<!--END--> |
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{{Knot Image Quadruple|L9n27|gif|L10n32|gif|L10n36|gif|L10n107|gif}} |
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{{Knot Image Quadruple|L11n244|gif|L11n247|gif|L11n334|gif|L11n381|gif}} |
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{{Knot Image Triple|L11n396|gif|L11n404|gif|L11n406|gif}} |
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[[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]]). |
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====Detecting a Link Using the Multivariable Alexander Polynomial==== |
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[[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: |
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<!--$$mva = MultivariableAlexander[L = PD[ |
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X[1, 16, 2, 17], X[3, 15, 4, 14], X[5, 8, 6, 9], |
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X[7, 21, 8, 20], X[9, 22, 10, 13], X[11, 2, 12, 3], |
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X[13, 18, 14, 19], X[15, 12, 16, 1], X[17, 11, 18, 10], |
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X[19, 4, 20, 5], X[21, 7, 22, 6] |
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]][t]$$--> |
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<!--Robot Land, no human edits to "END"--> |
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{{InOut| |
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n = 7 | |
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in = <nowiki>mva = MultivariableAlexander[L = PD[ |
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X[1, 16, 2, 17], X[3, 15, 4, 14], X[5, 8, 6, 9], |
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X[7, 21, 8, 20], X[9, 22, 10, 13], X[11, 2, 12, 3], |
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X[13, 18, 14, 19], X[15, 12, 16, 1], X[17, 11, 18, 10], |
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X[19, 4, 20, 5], X[21, 7, 22, 6] |
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]][t]</nowiki> | |
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out= <nowiki> 2 3 2 |
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-1 + 3 t[1] - 3 t[1] + t[1] + 3 t[2] - 7 t[1] t[2] + 7 t[1] t[2] - |
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3 2 2 2 2 |
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3 t[1] t[2] - 3 t[2] + 7 t[1] t[2] - 7 t[1] t[2] + |
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3 2 3 3 2 3 3 3 |
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3 t[1] t[2] + t[2] - 3 t[1] t[2] + 3 t[1] t[2] - t[1] t[2]</nowiki>}} |
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<!--END--> |
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<!--$$Select[AllLinks[], (MultivariableAlexander[#][t] == mva) &]$$--> |
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<!--Robot Land, no human edits to "END"--> |
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{{InOut| |
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n = 8 | |
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in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == mva) &]</nowiki> | |
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out= <nowiki>{Link[11, Alternating, 289]}</nowiki>}} |
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<!--END--> |
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And just to be sure, |
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<!--$${Jones[L][q], Jones[Link[11, Alternating, 289]][q]}$$--> |
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<!--Robot Land, no human edits to "END"--> |
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{{InOut| |
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n = 9 | |
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in = <nowiki>{Jones[L][q], Jones[Link[11, Alternating, 289]][q]}</nowiki> | |
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out= <nowiki> -(17/2) 4 8 12 16 18 17 15 |
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{q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + |
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15/2 13/2 11/2 9/2 7/2 5/2 3/2 |
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q q q q q q q |
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10 3/2 5/2 |
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------- - 7 Sqrt[q] + 3 q - q , |
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Sqrt[q] |
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-(5/2) 3 7 3/2 5/2 |
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-q + ---- - ------- + 10 Sqrt[q] - 15 q + 17 q - |
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3/2 Sqrt[q] |
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q |
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7/2 9/2 11/2 13/2 15/2 17/2 |
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18 q + 16 q - 12 q + 8 q - 4 q + q }</nowiki>}} |
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<!--END--> |
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Thus the mystery link is the mirror image of [[L11a289]]. |
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Revision as of 13:45, 28 August 2007
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(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[3]:=
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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[3]=
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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[4]:=
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mva - (mva /. {t1->t2, t2->t3, t3->t4, t4->t1})
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Out[4]=
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0
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In[5]:=
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mva - (mva /. {t1->t2, t2->t1})
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Out[5]=
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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.)
Links with Vanishing Multivariable Alexander Polynomial
There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is . Here they are:
