BraidClosure

The link a braid closes to — each out strand joined to the in slot below it, round the side. By Alexander's theorem every link is one, and a cell that evaluates it draws the closed braid.

WikipediaBraid theory↗

BraidClosure(braid)the closure of a braid, as a link

Typed signatures

Type	Package	Applies
(expression<Braid> | string) -> expression<Link>	braid	always

Domain: Braids and knots·Drawn as Graphics

Details

• Inert, like Braid: it is a value. The knot-polynomial heads (JonesPolynomial, AlexanderPolynomial, KauffmanBracket, BracketInvariant) take it and read the braid it closes
• Drawn as a StrandDiagram of the closure: the arcs join each out slot to the in slot below it, and a click follows a whole link component
• Wolfram Language has no counterpart. KnotData names knots by entity and returns braid properties of them, but no head closes a braid; neither a BraidData nor a BraidClosure is among Wolfram's system names. Not checked against a kernel (none on this box)

Examples

Basic #

The closure of $\sigma_1^3$ is the trefoil

live ↗

In$\mathrm{AlexanderPolynomial}(\mathrm{BraidClosure}(\mathrm{TorusBraid}(2, 3)))$Out$t^2-t+1$

About

Examples	1
Crosswalk	Wikipedia

See also: Braid, TorusBraid, TorusKnot, AlexanderPolynomial, JonesPolynomial