Smooth knots have compatible piecewise-linear exterior models

Assume countable choice. Let $K$ be a smooth oriented knot with a finite regular projection diagram $D$ having $n$ crossings and a supplied product tube. There is a polygonal knot $P$ inside that tube with the same crossing code as $D$ and a smooth rounding $K^{\mathrm{♯}}$ of $P$ such that $K^{\mathrm{♯}}$ is smoothly ambient isotopic to $K$. The compact exterior of this same $P$ has a good triangulation with at most $C(n+1)$ tetrahedra and a marked path for the original meridian with at most $C(n+1)$ boundary edges, for one universal constant $C$. The peripheral-preserving exterior homeomorphism transfers this model to $K^{\mathrm{♯}}$.

Oriented links in the three-sphere and ambient isotopy — smooth knot and ambient-isotopy convention.

Regular oriented link diagrams — finite regular diagram and crossing-code conventions.

Planar-diagram, Gauss, and signed Dowker–Thistlethwaite codes — finite regular diagram and crossing-code conventions.

Smooth-to-piecewise-linear knot exterior approximation — code-preserving polygonal approximation, smooth rounding and peripheral-preserving exterior homeomorphism.

Linear-size good exterior triangulation from a planar knot code — linear-size triangulation of the supplied polygonal realization and marked meridian.

Hass–Lagarias–Pippenger, §7 — diagram-to-exterior triangulation and linear-size bound.