AlexanderPolynomial

The Alexander polynomial of a knot, computed from the reduced Burau representation over $\mathbb{Z}[t,t^{-1}]$ and returned as an ordinary expression in $t$ . Takes the knot however it is named — as $T(p,q)$ , as a braid it closes from, or as a modular word.

WikipediaAlexander polynomialMathWorldAlexanderPolynomialWikidataQ1634206nLabAlexander polynomialBritannicatopic/Alexander-polynomial

AlexanderPolynomial(knot)

\Delta(t)

, normalised to start at

t^0

TorusKnot(p, q)

T(p,q)

as a knot — the closed form, with no braid involved

TwistKnot(n)the twist knot with

n

half-twists past its clasp — another closed form

PretzelKnot(p, q, r)

P(p,q,r)

as a knot, for odd

p,q,r

— closed form, no braid at all

FigureEightKnot()TwistKnot(1) under its own name

BurauMatrix(braid)the reduced Burau matrix itself

SeifertGenus(knot)

(p-1)(q-1)/2

for

T(p,q)

1

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

(c-s+1)/2

Typed signatures

Type	Package	Applies
`(expression<Braid> \| expression<FigureEightKnot> \| expression<Link> \| expression<PretzelKnot> \| expression<TorusKnot> \| expression<TwistKnot> \| string) -> expression`	`braid`	always

Domain: Braids and knots

Details

$\Delta(t) \doteq \det(\psi(\beta) - I)\cdot(1-t)/(1-t^n)$ , for $\psi$ the reduced Burau representation
$\Delta$ is only defined up to $\pm t^k$ , so results are normalised — lowest term at $t^0$ with a positive coefficient
$\Delta_{T(p,q)}(t) = (t^{pq}-1)(t-1)/((t^p-1)(t^q-1))$ , which is the oracle the Burau computation is checked against
$\Delta_{TwistKnot(n)}(t) = nt^2 - (2n+1)t + n$ ; $n=1$ is the figure-eight and $n=-1$ is the trefoil
$\Delta_{P(p,q,r)}(t) = \tfrac14[(pq+qr+rp)(t-2+t^{-1}) + (t+2+t^{-1})]$ , for odd $p,q,r$
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 $\mathbb{Z}[t,t^{-1}]$
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 $n$ or $(p,q,r)$ 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