Speculative Knots

Design: naming knots

Speculative design. Open, not settled — thinking and proposals, not shipped behavior. See Roadmap for what's in flight.

Split from Knots: §3 KnotData (deliberately not built), §4 loose ends.

KnotData, and why it is not this

Wolfram has no JonesPolynomial. It has KnotData["Trefoil", "JonesPolynomial"] — a lookup keyed by knot name, over the Rolfsen and Hoste–Thistlethwaite tables. That is a genuinely different facility from computing an invariant of a knot you constructed, and wanting one does not remove the need for the other:

• KnotData answers "what is known about the knot called $6_2$" from a table.
• JonesPolynomial(knot) answers "what is $V$ of the knot I just built" by computing.

A table would want: named knots (Trefoil, FigureEight, $8_{19}$), the standard properties (crossing number, genus, braid word, signature, unknotting number, symmetry type), and a way to go from a name to a Knot this package can compute with — which is the part that matters, because it is what lets the table cross-check the computation the way TorusKnot already does for torus knots.

The reason not to build it yet is that it is data, not design. Its value is in the table being right, and the table is large; the interesting question — what shape a knot takes in this package — is settled by §2 (Knots) without it.

Loose ends

• Knot families do not get heads of their own. TorusKnotCurve(p, q) became KnotCurve(knot, samples?) for the same reason TorusKnotJones went: the head names the operation, the argument names the knot. The *Curve suffix is the family — a head ending in Curve evaluates to a list of points (ParametricCurve, LorenzCurve, KnotCurve). LorenzCurve keeps its name because its argument is a step count, not a knot: it is a trajectory of an ODE that happens to draw knots, which is Ghys's point rather than a naming accident.
• Links are not knots. KauffmanBracket is defined for links and JonesPolynomial is not, which Knot does not currently express — the bracket simply takes a knot and works anyway when the closure has several components. A Link alongside Knot is the eventual answer; today the distinction lives in which head declines.