AlexanderPolynomial

The Alexander polynomial of a knot, computed from the reduced Burau representation over and returned as an ordinary expression in . Takes the knot however it is named — as , as a braid it closes from, or as a modular word.

WikipediaAlexander polynomial↗MathWorldAlexanderPolynomial↗WikidataQ1634206↗nLabAlexander polynomial↗Britannicatopic/Alexander-polynomial↗

AlexanderPolynomial(knot), normalised to start at

TorusKnot(p, q) as a knot — the closed form, with no braid involved

TwistKnot(n)the twist knot with half-twists past its clasp — another closed form

PretzelKnot(p, q, r) as a knot, for odd — closed form, no braid at all

FigureEightKnot()TwistKnot(1) under its own name

BurauMatrix(braid)the reduced Burau matrix itself

SeifertGenus(knot) for , for a twist or pretzel knot, else a POSITIVE braid's

Domain: Braids and knots

Details

• , for the reduced Burau representation
• is only defined up to , so results are normalised — lowest term at with a positive coefficient
• , which is the oracle the Burau computation is checked against
• ; is the figure-eight and is the trefoil
• , for odd
• A twist or pretzel knot is genus 1 always — the underlying Seifert surface has two disks joined by two or three bands, and adding a twist lengthens a band rather than adding one
• The determinant uses Bareiss elimination: every intermediate is a minor, so each division is exact over
• Bennequin: on a POSITIVE braid, Seifert's algorithm is already optimal, so the genus formula holds — on a mixed braid it does not
• No braid-word family in or is known for twist or pretzel knots in general, so `JonesPolynomial` and `KnotCurve` decline there except at the figure-eight and the trefoil, which carry the specific braid this package already had for them

Examples

See also: Braid, LorenzBraid