A Geometric Finiteness Theory for Essential Surfaces in Knot Exteriors

We develop a relative geometric finiteness theory for essential surfaces in knot exteriors. Let $\gamma$ be a unit-thickness representative of a knot type $K$, with $\operatorname{Len}(\gamma)\leq\Lambda$, and let $F\subset E(\gamma)$ be a properly embedded essential surface with $\operatorname{Area}(F)\leq\Delta$ and relative thickness at least $\tau$, defined using positive reach and controlled boundary collars. We prove that every bounded-geometry slice contains only finitely many pair-isotopy classes. We construct explicitly bounded canonical layered codes on a fixed ambient lattice and show that, at resolution $\varepsilon\leq c\min\{1,\tau\}$ with sufficiently fine angular quantization, equality of codes implies ambient pair-isotopy. Thus the topology of each bounded slice is recoverable from finite geometric data.
For a fixed exterior, these classes form finite visible subcomplexes of the essential-surface complex; the subcomplexes are monotone, exhaust the full complex, and carry isometric actions levelwise and meridian-preserving $C^{1,1}$ actions with controlled reindexing. Positive-reach compactness also yields attainment results for fixed-exterior and compactified visibility problems. Finally, the peripheral geometry gives a writhe window for connected surfaces with nonempty non-meridional boundary: \[
|r|\leq C_{\mathrm{BS}}\Lambda^{4/3}+w(\Delta,\tau). \] This produces slope invisibility gaps and a linear joint-area lower bound for a Seifert surface and cabling annulus of a torus knot. The framework is triangulation-free and complementary to normal-surface, branched-surface, sutured-manifold, and Heegaard-theoretic methods; it does not assert finiteness without geometric bounds.