The Determinant and the Signature: Difference between revisions
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{{Startup Note}} |
{{Startup Note}} |
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<!--$$?KnotDet$$--> |
<!--$$?KnotDet$$--> |
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{{HelpLine| |
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n = 2 | |
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in = <nowiki>KnotDet</nowiki> | |
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out= <nowiki>KnotDet[K] returns the determinant of a knot K.</nowiki>}} |
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<!--END--> |
<!--END--> |
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<!--$$?KnotSignature$$--> |
<!--$$?KnotSignature$$--> |
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<!--Robot Land, no human edits to "END"--> |
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{{HelpLine| |
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n = 3 | |
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in = <nowiki>KnotSignature</nowiki> | |
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out= <nowiki>KnotSignature[K] returns the signature of a knot K.</nowiki>}} |
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<!--END--> |
<!--END--> |
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Thus, for example, the knots [[5_1]] and [[10_132]] have the same determinant (and even the same Alexander and Jones polynomials), but different signatures: |
Thus, for example, the knots [[5_1]] and [[10_132]] have the same determinant (and even the same Alexander and Jones polynomials), but different signatures: |
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{{Knot Image Pair|5_1|gif|10_132|gif}} |
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<center>[[Image:5_1.gif|frame|none|<center>[[5_1]]</center>]] [[Image:10_132.gif|frame|none|<center>[[10_132]]</center>]]</center> |
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<!--$$KnotDet /@ {Knot[5, 1], Knot[10, 132]}$$--> |
<!--$$KnotDet /@ {Knot[5, 1], Knot[10, 132]}$$--> |
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{{InOut| |
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n = 4 | |
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in = <nowiki>KnotDet /@ {Knot[5, 1], Knot[10, 132]}</nowiki> | |
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out= <nowiki>{5, 5}</nowiki>}} |
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<!--END--> |
<!--END--> |
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<!--$${ |
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| ⚫ | |||
Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), |
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| ⚫ | |||
}$$--> |
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<!--Robot Land, no human edits to "END"--> |
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{{InOut| |
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n = 5 | |
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in = <nowiki>{ |
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Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), |
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Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]}) |
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}</nowiki> | |
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out= <nowiki>{True, True}</nowiki>}} |
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<!--END--> |
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<!--$$KnotSignature /@ {Knot[5, 1], Knot[10, 132]}$$--> |
<!--$$KnotSignature /@ {Knot[5, 1], Knot[10, 132]}$$--> |
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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>KnotSignature /@ {Knot[5, 1], Knot[10, 132]}</nowiki> | |
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out= <nowiki>{-4, 0}</nowiki>}} |
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<!--END--> |
<!--END--> |
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In August 2005 somebody emailed [[User:Drorbn|Dror]] a question about knot colouring, which amounted to "find the first knot (other than the unknot) whose determinant is <math>\pm 1</math>". So on September 2<sup>nd</sup> Dror typed |
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<!--$$Select[AllKnots[], Abs[KnotDet[#]] == 1 &]$$--> |
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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>Select[AllKnots[], Abs[KnotDet[#]] == 1 &]</nowiki> | |
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out= <nowiki>{Knot[0, 1], Knot[10, 124], Knot[10, 153], |
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Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42], |
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Knot[11, NonAlternating, 49], Knot[11, NonAlternating, 116]}</nowiki>}} |
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<!--END--> |
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Hence the first few knots that are not <math>k</math>-colourable for any <math>k</math> are [[10_124]], [[10_153]], [[K11n34]], [[K11n42]], [[K11n49]] and [[K11n116]]. |
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{{Knot Image|K11n116|gif}} |
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Latest revision as of 18:21, 21 February 2013
(For In[1] see Setup)
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Thus, for example, the knots 5_1 and 10_132 have the same determinant (and even the same Alexander and Jones polynomials), but different signatures:
 5_1 |
 10_132 |
In[4]:=
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KnotDet /@ {Knot[5, 1], Knot[10, 132]}
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Out[4]=
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{5, 5}
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In[5]:=
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{
Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}),
Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})
}
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Out[5]=
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{True, True}
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In[6]:=
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KnotSignature /@ {Knot[5, 1], Knot[10, 132]}
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Out[6]=
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{-4, 0}
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In August 2005 somebody emailed Dror a question about knot colouring, which amounted to "find the first knot (other than the unknot) whose determinant is ". So on September 2nd Dror typed
In[7]:=
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Select[AllKnots[], Abs[KnotDet[#]] == 1 &]
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Out[7]=
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{Knot[0, 1], Knot[10, 124], Knot[10, 153],
Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42],
Knot[11, NonAlternating, 49], Knot[11, NonAlternating, 116]}
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Hence the first few knots that are not -colourable for any are 10_124, 10_153, K11n34, K11n42, K11n49 and K11n116.
 K11n116 |
