The Determinant and the Signature

2009-05-22T11:37:36Z

EltouPassi:

http://www.textrobasvich.com
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{{Startup Note}}


{{HelpLine|
n = 1 |
in = <nowiki>KnotDet</nowiki> |
out= <nowiki>KnotDet[K] returns the determinant of a knot K.</nowiki>}}




{{HelpLine|
n = 2 |
in = <nowiki>KnotSignature</nowiki> |
out= <nowiki>KnotSignature[K] returns the signature of a knot K.</nowiki>}}


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}}



{{InOut|
n = 3 |
in = <nowiki>KnotDet /@ {Knot[5, 1], Knot[10, 132]}</nowiki> |
out= <nowiki>{5, 5}</nowiki>}}




{{InOut|
n = 4 |
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>}}




{{InOut|
n = 5 |
in = <nowiki>KnotSignature /@ {Knot[5, 1], Knot[10, 132]}</nowiki> |
out= <nowiki>{-4, 0}</nowiki>}}


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



{{InOut|
n = 6 |
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>}}


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}}

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The Determinant and the Signature