Ropelength-Filtered Essential Surface Spaces
Abstract
We develop a filtered geometric framework for essential surfaces in knot exteriors. Let γ be a unit-thickness representative of a knot type K with Len(γ)≤Λ, and let F be a properly embedded essential surface in the exterior of a fixed-radius tube about γ, with Area(F)≤Δ and relative thickness at least τ, 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 ε≤ cmin{1,τ} implies ambient pair-isotopy. In a fixed exterior E, the surface systems visible in a geometric window form finite subcomplexes ESΔ,τ(E) that exhaust the essential-surface complex; isometries act levelwise and C1,1 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|≤ CBSΛ4/3+w(Δ,τ). 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.
Related
- Ropelength-Filtered Reidemeister Graphs
- Ropelength-Filtered Swept-Area Geometry
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, III: the Spaces F∞,qα
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, II: Pointwise Multiplication, Localization, Differences
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups with Polynomial Growth, I: Inclusions, Discretization, Duality
BibTeX
@misc{ozawa2026ropelengthfiltered,
title = {{Ropelength-Filtered Essential Surface Spaces}},
author = {Makoto Ozawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.20844},
eprint = {2607.20844},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20844},
}
RIS
TY - UNPB
TI - Ropelength-Filtered Essential Surface Spaces
AU - Makoto Ozawa
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2607.20844
UR - https://arxiv.org/abs/2607.20844
AB - We develop a filtered geometric framework for essential surfaces in knot exteriors. Let γ be a unit-thickness representative of a knot type K with Len(γ)≤Λ, and let F be a properly embedded essential surface in the exterior of a fixed-radius tube about γ, with Area(F)≤Δ and relative thickness at least τ, 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 ε≤ cmin{1,τ} implies ambient pair-isotopy. In a fixed exterior E, the surface systems visible in a geometric window form finite subcomplexes ES_Δ,τ(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|≤ C_BSΛ^4/3+w(Δ,τ). 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.
ER -