The Determinant and the Signature: Difference between revisions

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<!--$$?KnotDet$$-->
<!--$$?KnotDet$$-->
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{{HelpLine|
{{Help1|n=1|s=KnotDet}}
n = 1 |
⚫
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>}}
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<!--$$?KnotSignature$$-->
<!--$$?KnotSignature$$-->
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{{HelpLine|
{{Help1|n=2|s=KnotSignature}}
n = 2 |
⚫
KnotSignature[K] returns the signature of a knot K.
in = <nowiki>KnotSignature</nowiki> |
{{Help2}}
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out= <nowiki>KnotSignature[K] returns the signature of a knot K.</nowiki>}}
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<!--$$KnotDet /@ {Knot[5, 1], Knot[10, 132]}$$-->
<!--$$KnotDet /@ {Knot[5, 1], Knot[10, 132]}$$-->
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{{InOut1|n=3}}
{{InOut|
n = 3 |
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<pre style="color: red; border: 0px; padding: 0em"><nowiki>KnotDet /@ {Knot[5, 1], Knot[10, 132]}</nowiki></pre>
{{InOut2|n=3}}<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}}
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<!--$${Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})}$$-->
<!--$${Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})}$$-->
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{{InOut1|n=4}}
{{InOut|
n = 4 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>{Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})}</nowiki></pre>
in = <nowiki>{Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})}</nowiki> |
{{InOut2|n=4}}<pre style="border: 0px; padding: 0em"><nowiki>{True, True}</nowiki></pre>
out= <nowiki>{True, True}</nowiki>}}
{{InOut3}}
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<!--$$KnotSignature /@ {Knot[5, 1], Knot[10, 132]}$$-->
<!--$$KnotSignature /@ {Knot[5, 1], Knot[10, 132]}$$-->
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{{InOut1|n=5}}
{{InOut|
n = 5 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>KnotSignature /@ {Knot[5, 1], Knot[10, 132]}</nowiki></pre>
⚫
in = <nowiki>KnotSignature /@ {Knot[5, 1], Knot[10, 132]}</nowiki> |
{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki>{-4, 0}</nowiki></pre>
out= <nowiki>{-4, 0}</nowiki>}}
{{InOut3}}
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Revision as of 14:00, 30 August 2005


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

(For In[1] see Setup)

In[1]:= ?KnotDet
KnotDet[K] returns the determinant of a knot K.
In[2]:= ?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
10_132
In[3]:= KnotDet /@ {Knot[5, 1], Knot[10, 132]}
Out[3]= {5, 5}
In[4]:= {Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})}
Out[4]= {True, True}
In[5]:= KnotSignature /@ {Knot[5, 1], Knot[10, 132]}
Out[5]= {-4, 0}
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