Main results

Let \(\mathcal K_N\) be the set of knot types with stick number at most \(N\). The paper proves

\[N^{(2/3+o(1))N} \leq |\mathcal K_N| \leq N^{3N+o(N)}.\]

Consequently

\[\log |\mathcal K_N|=\Theta(N\log N), \qquad |\mathcal K_N|=N^{\Theta(N)}.\]

This determines the order of growth of the logarithm, not an asymptotic equivalent or a limit for the normalized logarithm. The lower construction already produces prime satellite knots.

Proof architecture

An \(N\)-vertex polygon is represented in \(3N\) real coordinates. Its knot type can change only across a family of cubic and quartic self-intersection walls. A dimension-sensitive real-algebraic component bound, with the binomial denominator retained, gives the direct upper estimate \(N^{3N+o(N)}\).

The generic crossing-diagram route gives the coarser baseline \(\exp(O(N^2))\). A 2025 strong-geometry reconstruction theorem already implies the factorial scale; extracting the signs used in its proof gives \(N^{6N+o(N)}\). The direct discriminant count improves that extracted exponent from \(6\) to \(3\).

For the lower bound, an explicit family of doubly down–up permutations feeds long Garside words, endpoint purification, the Malyutin–Stupakov injection, and Huh–Oh’s arc-to-stick inequality. The resulting one-sided coefficient is \(2/3\).

The paper contains the complete proofs and citations. The formal-verification page records the exact machine-checked boundary.