SeifertGenus

The Seifert genus of a knot: the closed form for a named torus, twist or pretzel knot, otherwise Bennequin's $(c-s+1)/2$ on a positive braid's closure.

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) -> integer	braid	always

Domain: Braids and knots

Examples

Basic #

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

live ↗

In$\mathrm{SeifertGenus}(\mathrm{TorusKnot}(2, 5))$Out$2$

$(3-1)(4-1)/2$

live ↗

In$\mathrm{SeifertGenus}(\mathrm{TorusKnot}(3, 4))$Out$3$

About

Examples	2

See also: AlexanderPolynomial, BraidIsPositive