发表机构
Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University（驹泽大学文理学院自然科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究为纽结外部本质曲面建立几何有限性理论，通过构造有界规范分层码等方法，证明有界几何切片的同痕类有限，拓扑可由有限几何数据恢复，还得出扭结窗口等结果且框架无需三角剖分，补充了多种理论方法。
AI 中文摘要
我们为纽结外部的本质曲面发展了一种相对几何有限性理论。设γ是纽结类型K的单位厚度代表，其长度Len(γ)≤Λ，F⊂E(γ)是一个恰当嵌入的本质曲面，面积Area(F)≤Δ且相对厚度至少为τ，通过正到达和受控边界领圈定义。我们证明每个有界几何切片仅包含有限多个对同痕类。在固定环境格上明确构造有界规范分层码，并表明在分辨率ε≤c min{1,τ}且角度量化足够精细时，码的相等意味着环境对同痕。因此每个有界切片的拓扑可从有限几何数据恢复。对于固定外部，这些类形成本质曲面复形的有限可见子复形；子复形是单调的，耗尽整个复形，并携带等距作用和平行子午线的C¹,¹作用且索引受控。正到达紧致性也为固定外部和紧致化可见性问题产生达到结果。最后，周边几何为具有非空非子午线边界的连通曲面给出一个扭结窗口：|r|≤CBSΛ⁴/³ + w(Δ,τ)。这为环面纽结的赛弗特曲面和电缆环面产生斜率不可见间隙和线性联合面积下界。该框架无需三角剖分，是对正规曲面、分支曲面、缝合流形和希盖尔理论方法的补充；它在没有几何界时不主张有限性。
英文摘要
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 $\mathrm{Len}(γ)\leΛ$, and let $F$ be a properly embedded essential surface in the exterior of a fixed-radius tube about $γ$, with $\mathrm{Area}(F)\leΔ$ 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 $\varepsilon\le c\min\{1,τ\}$ 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 $|r|\le C_{\mathrm{BS}}Λ^{4/3}+w(Δ,τ)$. 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.
Comments109 pages, 21 figures, 2 tables
