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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{{Help1|n= |
{{Help1|n=1|s=KnotDet}} |
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KnotDet[K] returns the determinant of a knot K. |
KnotDet[K] returns the determinant of a knot K. |
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{{Help2}} |
{{Help2}} |
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<!--$$?KnotSignature$$--> |
<!--$$?KnotSignature$$--> |
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{{Help1|n= |
{{Help1|n=2|s=KnotSignature}} |
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KnotSignature[K] returns the signature of a knot K. |
KnotSignature[K] returns the signature of a knot K. |
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{{Help2}} |
{{Help2}} |
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<!--$$KnotDet /@ {Knot[5, 1], Knot[10, 132]}$$--> |
<!--$$KnotDet /@ {Knot[5, 1], Knot[10, 132]}$$--> |
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KnotDet /@ {Knot[5, 1], Knot[10, 132]} |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>KnotDet /@ {Knot[5, 1], Knot[10, 132]}</nowiki></pre> |
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{{InOut2|n= |
{{InOut2|n=3}}<pre style="border: 0px; padding: 0em"><nowiki>{5, 5}</nowiki></pre> |
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{{InOut3}} |
{{InOut3}} |
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<!--END--> |
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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= |
{{InOut1|n=4}} |
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{Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})} |
<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> |
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{{InOut2|n= |
{{InOut2|n=4}}<pre style="border: 0px; padding: 0em"><nowiki>{True, True}</nowiki></pre> |
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{{InOut3}} |
{{InOut3}} |
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<!--END--> |
<!--END--> |
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<!--$$KnotSignature /@ {Knot[5, 1], Knot[10, 132]}$$--> |
<!--$$KnotSignature /@ {Knot[5, 1], Knot[10, 132]}$$--> |
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{{InOut1|n= |
{{InOut1|n=5}} |
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KnotSignature /@ {Knot[5, 1], Knot[10, 132]} |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>KnotSignature /@ {Knot[5, 1], Knot[10, 132]}</nowiki></pre> |
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{{InOut2|n= |
{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki>{-4, 0}</nowiki></pre> |
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{{InOut3}} |
{{InOut3}} |
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<!--END--> |
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Revision as of 19:43, 27 August 2005
(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:
| In[3]:= |
KnotDet /@ {Knot[5, 1], Knot[10, 132]}
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| Out[3]= | {5, 5}
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| In[4]:= |
{Equal @@ (Jones[#][q]& /@ {Knot[5, 1], Knot[10, 132]}), Equal @@ (Alexander[#][t]& /@ {Knot[5, 1], Knot[10, 132]})}
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| Out[4]= | {True, True}
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| In[5]:= |
KnotSignature /@ {Knot[5, 1], Knot[10, 132]}
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| Out[5]= | {-4, 0}
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