The Alexander-Conway Polynomial

2009-05-21T19:55:06Z

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{{Startup Note}}



{{HelpAndAbout|
n = 1 |
n1 = 2 |
in = <nowiki>Alexander</nowiki> |
out= <nowiki>Alexander[K][t] computes the Alexander polynomial of a knot K as a function of the variable t. Alexander[K, r][t] computes a basis of the r'th Alexander ideal of K in Z[t].</nowiki> |
about= <nowiki>The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.</nowiki>}}




{{HelpLine|
n = 3 |
in = <nowiki>Conway</nowiki> |
out= <nowiki>Conway[K][z] computes the Conway polynomial of a knot K as a function of the variable z.</nowiki>}}


{{Knot Image|8_18|gif}}

The Alexander polynomial <math>A(K)</math> and the Conway polynomial <math>C(K)</math> of a knot <math>K</math> always satisfy <math>A(K)(t)=C(K)(\sqrt{t}-1/\sqrt{t})</math>. Let us verify this relation for the knot [[8_18]]:



{{InOut|
n = 4 |
in = <nowiki>alex = Alexander[Knot[8, 18]][t]</nowiki> |
out= <nowiki> -3 5 10 2 3
13 - t + -- - -- - 10 t + 5 t - t
2 t
t</nowiki>}}




{{InOut|
n = 5 |
in = <nowiki>Expand[Conway[Knot[8, 18]][Sqrt[t] - 1/Sqrt[t]]]</nowiki> |
out= <nowiki> -3 5 10 2 3
13 - t + -- - -- - 10 t + 5 t - t
2 t
t</nowiki>}}


The ''determinant'' of a knot <math>K</math> is <math>|A(K)(-1)|</math>. Hence for [[8_18]] it is



{{InOut|
n = 6 |
in = <nowiki>Abs[alex /. t -> -1]</nowiki> |
out= <nowiki>45</nowiki>}}


Alternatively (see [[The Determinant and the Signature]]):



{{InOut|
n = 7 |
in = <nowiki>KnotDet[Knot[8, 18]]</nowiki> |
out= <nowiki>45</nowiki>}}


<math>V_2(K)</math>, the (standardly normalized) type 2 Vassiliev invariant of a knot <math>K</math> is the coefficient of <math>z^2</math> in its Conway polynomial:



{{InOut|
n = 8 |
in = <nowiki>Coefficient[Conway[Knot[8, 18]][z], z^2]</nowiki> |
out= <nowiki>1</nowiki>}}


Alternatively (see [[Finite Type (Vassiliev) Invariants]]),



{{InOut|
n = 9 |
in = <nowiki>Vassiliev[2][Knot[8, 18]]</nowiki> |
out= <nowiki>1</nowiki>}}


{{Knot Image Pair|K11a99|gif|K11a277|gif}}

Sometimes two knots have the same Alexander polynomial but different Alexander ideals. An example is the pair [[K11a99]] and [[K11a277]]. They have the same Alexander polynomial, but the second Alexander ideal of the first knot is the whole ring <math>{\mathbb Z}[t]</math> while the second Alexander ideal of the second knot is the smaller ideal generated by <math>3</math> and by <math>1+t</math>:



{{In|
n = 10 |
in = <nowiki>{K1, K2} = {Knot[11, Alternating, 99], Knot[11, Alternating, 277]};</nowiki>}}




{{InOut|
n = 11 |
in = <nowiki>Alexander[K1] == Alexander[K2]</nowiki> |
out= <nowiki>True</nowiki>}}




{{InOut|
n = 12 |
in = <nowiki>Alexander[K1, 2][t]</nowiki> |
out= <nowiki>{1}</nowiki>}}




{{InOut|
n = 13 |
in = <nowiki>Alexander[K2, 2][t]</nowiki> |
out= <nowiki>{3, 1 + t}</nowiki>}}


Finally, the Alexander polynomial attains 551 values on the 802 knots known to <code>KnotTheory`</code>:



{{InOut|
n = 14 |
in = <nowiki>Length /@ {Union[Alexander[#]& /@ AllKnots[]], AllKnots[]}</nowiki> |
out= <nowiki>{551, 802}</nowiki>}}

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The Alexander-Conway Polynomial