All knots, such as this torus knot,

KnotData[{"TorusKnot", {3, 5}}]

Torus knot

have an associated Alexander polynomial:

KnotData[{"TorusKnot", {3, 5}}, "AlexanderPolynomial"][x]

$$x^4+\frac{1}{x^4}-x^3-\frac{1}{x^3}+x+\frac{1}{x}-1 .$$

Why aren't the Alexander polynomials for a pretzel knot, such as this,

Pretzel knot


KnotData[{"PretzelKnot", {5, 4, 3}}, "AlexanderPolynomial"]

  • 2
    $\begingroup$ I guess that the properties in KnotData are not computed on the fly but that they are tabulated. I am not sure; maybe the "KnotTheory`" package can help you further. $\endgroup$ – Henrik Schumacher Dec 15 '18 at 7:14
  • $\begingroup$ Supporting Henrik's supposition, comparing Length@KnotData[{"PretzelKnot", {5, 4, 3}}, "Properties"] (i.e., 67) with Length@DeleteCases[KnotData[{"PretzelKnot", {5, 4, 3}}, #]& /@ KnotData[{"PretzelKnot", {5, 4, 3}}, "Properties"], Missing["NotAvailable"]] (i.e., 15), you see that most of the knot properties are not available for this particular knot. $\endgroup$ – Bob Hanlon Dec 15 '18 at 15:37

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