The Determinant and the Signature: Difference between revisions

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{{Startup Note}}
{{Startup Note}}
<!--$$?KnotDet$$-->
<!--$$?KnotDet$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{HelpLine|
{{Help1|n=2|s=KnotDet}}
n = 2 |
⚫
KnotDet[K] returns the determinant of a knot K.
in = <nowiki>KnotDet</nowiki> |
{{Help2}}
⚫
out= <nowiki>KnotDet[K] returns the determinant of a knot K.</nowiki>}}
<!--END-->
<!--END-->


<!--$$?KnotSignature$$-->
<!--$$?KnotSignature$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{HelpLine|
{{Help1|n=3|s=KnotSignature}}
n = 3 |
⚫
KnotSignature[K] returns the signature of a knot K.
in = <nowiki>KnotSignature</nowiki> |
{{Help2}}
⚫
out= <nowiki>KnotSignature[K] returns the signature of a knot K.</nowiki>}}
<!--END-->
<!--END-->


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:


{{Knot Image Pair|5_1|gif|10_132|gif}}
{| align=center
|[[Image:5_1.gif|thumb|80px|<center>[[5_1]]</center>]]
|[[Image:10_132.gif|frame|none|<center>[[10_132]]</center>|120px]]
|}


<!--$$KnotDet /@ {Knot[5, 1], Knot[10, 132]}$$-->
<!--$$KnotDet /@ {Knot[5, 1], Knot[10, 132]}$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{InOut1|n=4}}
{{InOut|
n = 4 |
⚫
KnotDet /@ {Knot[5, 1], Knot[10, 132]}
{{InOut2|n=4}}<pre style="border: 0px; padding: 0em"><nowiki>{5, 5}</nowiki></pre>
in = <nowiki>KnotDet /@ {Knot[5, 1], Knot[10, 132]}</nowiki> |
out= <nowiki>{5, 5}</nowiki>}}
{{InOut3}}
<!--END-->
<!--END-->


<!--$${
⚫
<!--$${Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})}$$-->
⚫
Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}),
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
⚫
Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})
{{InOut1|n=5}}
}$$-->
⚫
{Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})}
<!--Robot Land, no human edits to "END"-->
{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki>{True, True}</nowiki></pre>
{{InOut|
{{InOut3}}
n = 5 |
in = <nowiki>{
⚫
Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}),
⚫
Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})
}</nowiki> |
out= <nowiki>{True, True}</nowiki>}}
<!--END-->
<!--END-->


<!--$$KnotSignature /@ {Knot[5, 1], Knot[10, 132]}$$-->
<!--$$KnotSignature /@ {Knot[5, 1], Knot[10, 132]}$$-->
<!--Robot Land, no human edits to "END"-->
<!--The lines to END were generated by WikiSplice: do not edit; see manual.-->
{{InOut1|n=6}}
{{InOut|
n = 6 |
⚫
KnotSignature /@ {Knot[5, 1], Knot[10, 132]}
in = <nowiki>KnotSignature /@ {Knot[5, 1], Knot[10, 132]}</nowiki> |
{{InOut2|n=6}}<pre style="border: 0px; padding: 0em"><nowiki>{-4, 0}</nowiki></pre>
out= <nowiki>{-4, 0}</nowiki>}}
{{InOut3}}
<!--END-->
<!--END-->

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

<!--$$Select[AllKnots[], Abs[KnotDet[#]] == 1 &]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 7 |
in = <nowiki>Select[AllKnots[], Abs[KnotDet[#]] == 1 &]</nowiki> |
out= <nowiki>{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]}</nowiki>}}
<!--END-->

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]].

{{Knot Image|K11n116|gif}}

Latest revision as of 17:21, 21 February 2013


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

(For In[1] see Setup)

In[2]:= ?KnotDet
KnotDet[K] returns the determinant of a knot K.
In[3]:= ?KnotSignature
KnotSignature[K] returns the signature of a knot K.

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.gif
5_1 		10 132.gif
10_132
In[4]:= KnotDet /@ {Knot[5, 1], Knot[10, 132]}
Out[4]= {5, 5}
In[5]:= { Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]}) }
Out[5]= {True, True}
In[6]:= KnotSignature /@ {Knot[5, 1], Knot[10, 132]}
Out[6]= {-4, 0}

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]:= Select[AllKnots[], Abs[KnotDet[#]] == 1 &]
Out[7]= {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]}

Hence the first few knots that are not -colourable for any are 10_124, 10_153, K11n34, K11n42, K11n49 and K11n116.

K11n116.gif
K11n116
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