Ropelength-Filtered Essential Surface Spaces

Abstract

We develop a filtered geometric framework for essential surfaces in knot exteriors. Let $\gamma$ be a unit-thickness representative of a knot type $K$ with $\mathrm{Len}(\gamma )\le \Lambda$, and let $F$ be a properly embedded essential surface in the exterior of a fixed-radius tube about $\gamma$, with $\mathrm{Area}(F)\le \Delta$ and relative thickness at least $\tau$, formulated through Federer reach and controlled boundary collars. We prove that this bounded-geometry pair space contains only finitely many pair-isotopy classes, and that equality of explicitly bounded canonical layered codes at resolution $\epsilon \le c\min \{1,\tau \}$ implies ambient pair-isotopy. In a fixed exterior $E$, the surface systems visible in a geometric window form finite subcomplexes ${ES}_{\Delta ,\tau }(E)$ that exhaust the essential-surface complex; isometries act levelwise and $C^{1,1}$ self-diffeomorphisms act with controlled reindexing. For a fixed two-sided surface, compressing disks are filtered in the same way, giving finite geometric witnesses for compressibility and weak reducibility and recovering the index-one characterization in Bachman's topological index theory. On the peripheral torus, every visible numerical boundary slope lies in an explicit writhe window, so that $|r|\le C_{\mathrm{BS}}{\Lambda }^{4/3}+w(\Delta ,\tau )$. Together with finite Reidemeister certificates, the faithful codes give two independent finite recognition mechanisms, one for the knot and one for the carried essential-surface type. The framework is a smooth, triangulation-free analogue of the finiteness philosophy of normal surface theory; it does not assert that a knot exterior has only finitely many essential surfaces without geometric bounds.

arXiv comment109 pages, 21 figures, 2 tables