arXiv subjects

Makoto Ozawa

Publications and source records attributed to Makoto Ozawa.

At least 19 recordsLinked to original sources

arXiv2026

Ropelength-Filtered Essential Surface Spaces

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.

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arXiv2026

Ropelength-Filtered Knot Densities

We study scale-invariant $p$-densities of knot types in Euclidean three-space, defined as the infimum over representatives of the length divided by an $L^p$-type mean of the pairwise chord lengths; the exponent $p$ is any value greater than minus one, including infinity, where the mean is the diameter. The unconstrained density is independent of the knot type. For exponents at most two, its common value is the density of the round circle, by the sharp mean-chord inequality; at the diameter endpoint it equals two; and for finite exponents greater than two it reduces to a planar convex extremal problem, in which the round circle fails to be extremal beyond an explicit threshold. To prevent this degeneration, which is caused by knotting at arbitrarily small scale, we introduce a ropelength filtration: at each level $λ$, the infimum is restricted to representatives whose ropelength is at most $λ$ times the ropelength of the knot type. The filtered density has minimizers, detects the unknot for exponents at most two, and converges to the unconstrained density as $λ$ tends to infinity. It is continuous in the exponent, including the diameter endpoint, and right-continuous in the filtration parameter. As a numerical test, we evaluate the density on published approximately tight configurations of the granny knot and the square knot. The sampled values differ by about 2.5 percent at exponent two, whereas their ordering is reversed at the diameter endpoint. These computations provide evidence that the ropelength filtration retains geometric information lost by the unconstrained theory.

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arXiv2026

Ropelength-Filtered Swept-Area Geometry

This paper studies the swept-area cost of isotopies between thick knot representatives when the isotopy is required to stay inside a ropelength window: every intermediate curve has thickness at least one and length at most $Λ$. Without these constraints the swept area is the classical homotopy-area length on spaces of curves, and the induced distance on unparametrized curves is bounded below by the flat norm; see Yezzi--Mennucci and Michor--Mumford. We record the corresponding non-degeneracy on the ropelength-filtered moduli space, taken modulo orientation-preserving reparametrizations and Euclidean isometries, as a consequence of this classical lower bound. The ropelength window changes the theory in two ways. Distances are infinite between classes that are not yet connected at the level $Λ$, and all costs depend monotonically on the budget. We organize this dependence through budget--cost profiles and swept-area merge costs of admissible components. We prove their monotonicity and a transport estimate under whole-path simulations, and relate them to the merge scales of ropelength-filtered knot spaces. On the diagrammatic side, we construct a network with exact spatial endpoints whose path cost equals the infimal cost over diagrammatically generic isotopies, and show that the graph obtained by collapsing projection fibres gives only a lower bound, which can lose positive cost inside a fibre. We also give projected-area calibrations, exact formulas for concentric round and homothetic elliptical unknots, in which the window is not active, a labelled polygonal estimate, and a based loop-length structure on admissible fundamental groups. Existence of minimizing isotopies under the thickness and length constraints is left open.

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arXiv2026

Merge Trees of Length-Filtered Lattice Knot Spaces

We study lattice-filtered move graphs as finite-state models for knot types under a length cap. At level $N$, vertices are lattice polygons of a fixed knot type with length at most $N$, modulo orientation-preserving lattice isometries, and edges are local moves. The first level at which two initial components become connected defines a discrete merge scale; after subtracting the birth level it is an ultrapseudometric. For the standard BFACF moves on the simple cubic lattice, the theorem of Janse van Rensburg and Whittington gives connectivity without a cap; our question is the least cap connecting a prescribed pair, and explicit BFACF paths serve as finite PL isotopy certificates. We completely determine the minimal-layer merge trees of the amphichiral knots $4_1$ and $6_3$. The $152$ minimal $4_1$ classes form four components of sizes $58,58,18,18$ at $N=30$ and a single component at $N=32$, so the merge tree is $4\to1$ with barrier $2$. The $148$ minimal $6_3$ classes form twelve components at $N=40$, exchanged in pairs by reflection; at $N=42$ they merge into two mirror components, each with $74$ minimal classes and $12337$ states, and a verified path joins them at $N=44$. Hence the merge tree is $12\to2\to1$ with possible barriers $0,2,4$. Independently verified seed-to-mirror certificates realize the extremal barriers $2$ for $4_1$ and $4$ for $6_3$. Checks for the trefoil, the five-crossing prime knots and composite trefoils are included as reproducibility tests.

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arXiv2026

Ropelength-Filtered Reidemeister Graphs

We prove relative smoothing with uniform reach control for one-parameter families of positive-reach $C^{1,1}$ knots and combine it with relative multijet transversality to obtain projection-generic regularization inside strict ropelength sublevels. For families over a compact polyhedron we prove relative smoothing of the slices after an arbitrarily small budget increase. Passing to a lifted Reidemeister multigraph reconstructs the admissible $π_0$ of every strict projection-framed ropelength sublevel. We also compare the strong, quotient, and uniform-slack path relations and show that the corresponding path and admissible merge thresholds agree. The normalised projection-framed sublevels are direction-marked spaces over the normalised ropelength sublevels of arXiv:2604.17905; in particular they are compact Hausdorff, so connected merge levels are attained. At closed critical levels we use only arbitrarily small right relaxation and do not assert exact genericity within the critical level itself. For finite recognition we distinguish three notions that play different roles: full rooted-ball occurrence in the completed $S^2$-Reidemeister multigraph, visibility in the monotone diagram-image filtration, and saturation of a visible occurrence. Every finite Reidemeister submultigraph becomes visible at a finite ropelength level, and a visible saturated BC-characteristic certificate determines the knot type up to mirroring. A universal computable crossing-complexity bound controls visibility. Finally, vertex-coherent lifting of a finite pattern is characterized by a face-consistent planar labeling, in particular for tree-shaped patterns and separated cube systems.

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arXiv2026

Ropelength-Filtered Knot Spaces

Ropelength is usually studied as a minimisation problem for one knot at a time. Here we instead use ropelength to filter the space of all realisations of a knot type. For a knot type $K$ and budget $Λ$, let $Y_Λ(K)$ be the space of unit-thickness $C^{1,1}$ configurations of length at most $Λ$, modulo rigid motions and constant-speed reparametrisation. We prove compact capture for smooth knot families, yielding homotopical and homological exhaustion of the ordinary knot space by finite ropelength levels. In the normalised Euclidean model, every $Y_Λ(K)$ is compact; connected components are right-continuous and their merge levels are attained, giving an ultrametric on the components of the ideal stratum. We also prove that $Y_L(K)$ strongly deformation retracts onto its exact-length shell, so path connectivity at a fixed level is fixed-length physical isotopy. The connected merge scale and the path merge scale, the min--max quantity for constrained deformation paths, take the same values, although only the former is known to be attained. Moreover, all configurations in any fixed finite sublevel become path connected to one another at some common finite higher level, even if the original sublevel has infinitely many components. For higher homotopy we introduce ropelength widths and relate them to known models of spaces of knots. We record mirror symmetry and prove finite factorization across every positive filtration gap in a marked weak model, yielding all-degree homological q-tameness. Finally, we relate the path filtration to the companion swept-area construction: the infimal budget for admissible endpoint connections agrees with the path merge threshold. The resulting framework separates geometric, topological, and physical-isotopy questions and provides a quantitative foundation for the study of ropelength-filtered knot spaces.

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arXiv2026

Density and Compression on Lattice-Knot Merge Trees

Ropelength is usually studied as a minimization problem for one knot at a time. We instead use ropelength to filter the space of all realizations of a knot type. For a knot type $K$ and a budget $Λ$, let $Y_Λ(K)$ be the space of unit-thickness $C^{1,1}$ configurations of length at most $Λ$, modulo rigid motions and constant-speed reparametrization. We prove compact capture for smooth knot families, yielding homotopical and homological exhaustion of the ordinary knot space by finite ropelength levels. In the normalized Euclidean model we prove compactness of every $Y_Λ(K)$, right continuity of connected components, and attainment of connected merge levels; on components of the ideal stratum these levels define an ultrametric. We also prove that $Y_L(K)$ strongly deformation retracts onto its exact-length shell. This separates an attained connected merge scale from the path merge scale, the genuine min-max quantity for constrained deformation paths. For higher homotopy we introduce ropelength widths and relate them to known models of spaces of knots. We record mirror symmetry and the complete minimal-layer BFACF merge trees of $4_1$ and $6_3$ as certified discrete benchmarks, without identifying discrete and continuum barriers. Finally, we relate the path filtration to the companion swept-area construction: at fixed ropelength its infimal trace area defines a genuine extended metric on the corresponding moduli space, with flat-current and projected-area lower bounds. The paper also formulates open problems concerning Gordian phenomena, regularization, and quantitative topology of thick-knot spaces.

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arXiv2026

Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction

The curve complex of a surface is deliberately forgetful, yet in most cases its automorphism group recovers the mapping class group. We develop an analogous image--kernel--reconstruction viewpoint for the essential-surface complex ${ES}(E)$ of a knot exterior $E=E(K)$, equivalently Schultens's initial surface complex $S_0(E)$. The paper is written as an entry point, beginning with explicit classical examples before introducing the general framework. We show that ${ES}(E)$ is a flag complex, that its boundary-bearing vertices are layered by boundary slope, and that although each ${ES}(E)$ is finite-dimensional, its dimension is unbounded over all knots. Kakimizu complexes of incompressible and minimal-genus spanning surfaces occur naturally as full subcomplexes. For slope-separated knots we determine the natural mapping-class-group action: every orientation-preserving mapping class acts trivially, while a nontrivial image occurs precisely through amphichirality. We illustrate the theory with torus knots, the figure-eight knot, cable knots, and connected sums; in the latter case annular spinning is already visible on ${ES}(E)$. We conclude with a recognition--realization--kernel roadmap, graded problems, and a worked two-bridge-knot recipe.

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arXiv2026

Amicable Knots on Minimal Genus Seifert Surfaces

For a knot $K\subset S^3$, let $S(K)$ denote the set of non-trivial knot types represented by simple closed curves on a minimal genus Seifert surface of $K$. We study the relation $J\in S(K)$ and its symmetric part, which leads to the notion of \emph{amicable knots}: knots $K$ and $J$ are called amicable if each is represented by a simple closed curve on a minimal genus Seifert surface of the other. A classical result of Lyon implies that the family of torus knots is universal for this realization problem: for every non-trivial knot type $J$, there exists a torus knot $T$ such that $J\in S(T)$. In contrast, one of the main results of this paper is that no single knot is universal: for every knot $K$, there exists a knot $J$ such that $J\notin S(K)$. We also study explicit examples, keeping track of chirality throughout. Writing $3_1^+=T(2,3)$ and $8_{19}^+=T(3,4)$ for the right-handed positive torus knots, we show that $3_1^+$ and $8_{19}^+$ are amicable, whereas $3_1^+$ and the figure-eight knot $4_1$ are not. We also describe the hosting sets of both chiralities of the trefoil in terms of primitive slope classes on their once-punctured torus fibers.

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arXiv2026

Geometric densities and compression radii of knot types

We introduce scale-free compression radii and packing ratios of knot types and clarify their relation to geometric densities. Let $D$ be a Euclidean-invariant, scale-covariant size functional on embedded closed curves. For a curve $γ$, we define the $D$-density by $\operatorname{Len}(γ)/D(γ)$, the $D$-compression radius by $D(γ)/\operatorname{Thi}(γ)$, and the corresponding packing ratio as its reciprocal. For each representative, ropelength is the product of the $D$-density and the $D$-compression radius. The main point is not this formal cancellation, but the separation it suggests after optimization within a fixed knot type: density, compression, and ropelength generally have different minimizing sequences. We establish the basic optimized inequality and a criterion for equality, and compute the unknot case for diameter and minimal enclosing radius. We also prove polygonal approximation theorems for the compression radii associated with these two size functionals, using standard convergence properties of polygonal thickness, and formulate sufficient hypotheses for analogous results for other $L^p$-type size functionals. Finally, we discuss relations with distortion, trunk, and supertrunk. The framework is intended as a structural companion to density-type invariants rather than as an immediate source of stronger ropelength lower bounds. In particular, the optimized factorization alone does not yield new ropelength bounds; such bounds require independent estimates for the density and compression factors.

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arXiv2026

Crossing Numbers of Knots on Closed Surfaces

Let c(K;F) denote the surface crossing number of a knot K with respect to a closed connected surface F in S^3. We relate c(K;F) to the tunnel number t(K) and to the Heegaard deficiency delta(F)=g(M_1;F)+g(M_2;F)-g(F), where S^3=M_1 union_F M_2. The zero-crossing case gives a structural obstruction: if c(K;F)=0, then t(K) <= delta(F). Conversely, if t(K)>delta(F), then c(K;F) >= 2(t(K)-delta(F))+1. Thus the Heegaard deficiency of F measures the amount of tunnel complexity that can be absorbed by F without producing crossings. The proof combines a surface ascending-number estimate, a bridge-number estimate for surface diagrams, and an amalgamation argument for Heegaard splittings relative to F. We also construct connected-sum families showing that the lower bound has the correct linear order.

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arXiv2026

A Kuratowski-Type Classification of Critical Complexes for the 3-Sphere

We give a Kuratowski-type classification of a graph-defined class of minimal piecewise-linear obstructions to embeddability in the 3-sphere. A finite simplicial complex $X$ is called critical for $S^3$ if $|X|$ does not embed in $S^3$, whereas deleting the open star of any simplex in the second barycentric subdivision of $X$ yields a polyhedron embeddable in $S^3$. The main theorem completely classifies critical complexes of the form $(G\times S^1)\cup H$, where $G$ and $H$ are graphs and $H$ is attached along vertices of the branch set of $G\times S^1$. We prove that there are exactly seven such complexes up to homeomorphism: two $K_4$-type complexes, four $Θ_4$-type complexes, and one $K_{2,3}$-type complex. The proof is combinatorial in nature. By collapsing the $S^1$-factor of $G\times S^1$, we associate to $X$ a reduction graph $\widehat X=G\cup H$. Criticality implies that $H$ is a forest, $G$ is planar, and $\widehat X$ is inclusion-minimal non-planar. Kuratowski's theorem therefore reduces the classification to the cases $K_5$ and $K_{3,3}$. A finite analysis of forest attachments, together with a face-incidence criterion for embeddability, leaves precisely the seven models listed above. We also prove that every non-embeddable regular multibranched surface in $S^3$ contains a critical subcomplex of the form $M\cup H$, where $M$ is a regular multibranched surface and $H$ is a graph.

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arXiv2026

A Dehornoy-Type Ordering on Plat Presentation Classes

For each integer $n\ge 1$, after fixing a proper complexity function on the braid group $\B_{2n}$, we use the Dehornoy order to define a strict total order on the set \[ \mathcal P_{2n}=H_{2n}\backslash \B_{2n}/H_{2n} \] of $2n$--plat presentation classes. For a link type $\mathcal L$ with bridge number $b(\mathcal L)\le n$, this induces a strict total order on the subset $\mathcal P^{(n)}(\mathcal L)$ corresponding to bridge isotopy classes of $n$--bridge positions of $\mathcal L$. We also define a distinguished class $\CanPlat_D^{(n)}(\mathcal L)$ and show that the globally chosen Dehornoy canonical braid agrees with the cosetwise canonical representative of the associated Hilden double coset. As an application, we reformulate the fixed-level bridge finiteness conjecture in terms of boundedness of canonical representatives. This viewpoint supports the role of bridge positions as a structured finite-level model for studying the otherwise vast collection of geometric positions of a link.

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arXiv2026

3-decompositions of genus two handlebody-knots

We investigate the class of $3$-decomposable genus two handlebody-knots and provide a complete classification of essential annuli in their exteriors. We introduce the notion of $τ$- and $ρ$-tangles and good rectangles and annuli. By classifying $τ$- and $ρ$-tangles whose exteriors admit a good rectangle or annulus, we categorize atoroidal $3$-decomposable genus two handlebody-knots into distinct classes, based on the number of essential annuli. As an application, the hyperbolicity of all genus two handlebody-knots with up to six crossings are determined, and numerous hyperbolic handlehody-knots with seven crossings identified. Furthermore, obstructions for a handlebody-knot to be $3$-decomposable are constructed with explicit examples provided.

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arXiv2026

The realization problem of essential surfaces in knot exteriors

We study compact orientable essential surfaces in knot exteriors in the 3-sphere. The genus $g$, the number of boundary components $b$, and the boundary slope $p/q$ are fundamental invariants of an essential surface. The \textit{realization problem} asks whether, for a given triple $(g, b, q)$ with $g \ge 0$, $b \ge 1$, and $q \ge 1$, there exists a knot $K \subset S^3$ whose exterior $E(K)$ contains a compact orientable essential surface $F$ of genus $g$ with $b$ boundary components and boundary slope $p/q$ for some $p$. In general, not all combinations of $(g, b, q)$ are realizable. First, we show that if $b$ is odd, then $q$ must be equal to $1$. Our main theorem states that for any given even $b \ge 2$ and $q \ge 1$, there exist a genus $g \ge 0$ and a knot $K$ such that $E(K)$ contains a compact orientable essential surface with these parameters.

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arXiv2026

Segment number of knots

We introduce a new numerical knot invariant, termed the \textit{segment number}, which is derived from partitioned knot diagrams subject to specific over/under-crossing constraints. We prove that a knot is non-trivial if and only if its segment number is at least 3. Furthermore, we investigate the structural properties of the directed graph associated with a minimal segment number presentation. Specifically, we show that for any minimal presentation, the underlying graph is connected and cannot be a path. Finally, we discuss the relationship between the segment number and the bridge number, providing bounds and conjectures for future study. We also conjecture that the bridge number $b(K)$ provides a lower bound for the segment number.

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arXiv2025

On symmetry and exterior problems of knotted handlebodies

The paper concerns two classical problems in knot theory pertaining to knot symmetry and knot exteriors. In the context of a knotted handlebody $V$ in a $3$-sphere $S^3$, the symmetry problem seeks to classify the mapping class group of the pair $(S^3,V)$, whereas the exterior problem examines to what extent the exterior $E(V)$ determines or fails to determine the isotopy type of $V$. The paper determines the symmetries of knotted genus two handlebodies arising from hyperbolic knots with non-integral toroidal Dehn surgeries, and solve the knot exterior problem for them. A new interpretation and generalization of a Lee-Lee family of knotted handlebodies is provided.

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arXiv2024

Essential annuli in genus two handlebody exteriors

We classify all potential configurations of essential annuli in a genus two atoroidal handlebody exterior in the $3$-sphere, building on two recent classifications: the classification of the JSJ-graph of the exterior and the classification of essential annuli in the exterior. In contrast to knots, genus two handlebody exteriors may contain infinitely many non-isotopic essential annuli, due to the JSJ-graph classification. Our main result characterizes the numbers of different types of essential annuli in such an infinite family.

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