Alexander Polynomial

Author

Eric W. Weisstein
August 9, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/KnotTheory/AlexanderPolynomial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/AlexanderPolynomial.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Knots`Knots`

Examples

{#,AlexanderDelta[#,x],ConwayNabla[#,x]}&/@
{TrefoilKnot,FigureEightKnot,SolomonsSealKnot,StevedoresKnot,MillerInstituteKnot}//TableForm//TraditionalForm

Links

AlexanderDelta[{11,-9,3},x]//TraditionalForm

AlexanderDelta[{{-2,6,-4,1},{1,-4,6,-2}},{x,y}]//TraditionalForm

AlexanderDelta[{{0,1,-1},{1,-2,2},{-1,2,-1}},{x,y,z}]//TraditionalForm

AlexanderDelta[{{{-1,1},{0,-1}},{{2,-1},{-1,1}}},{x,y,z,w}]//TraditionalForm

Δ(1)

AlexanderDelta[#,1]&/@Knots

Δ(-1)

Mod[AlexanderDelta[#,-1]&/@Knots,8]

Table

Alexander-Conway

{#,ConwayNabla[#,t]}&/@Knots//TableForm//TraditionalForm

Alexander

{#,AlexanderDelta[#,t]}&/@Knots//TableForm//TraditionalForm

Normalized a la Livingston

AlexanderDeltaNormalized[link_Link,x_]:=Module[
	{delta=Numerator[Together[AlexanderDelta[link,x]]]},
	Sign[First[CoefficientList[#,x]]]#&[delta]
]

{#,AlexanderDeltaNormalized[#,t]}&/@Knots//TableForm//TraditionalForm