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Osamu Saeki

Publications and source records attributed to Osamu Saeki.

At least 19 recordsLinked to original sources
arXiv2026

An Improved FPT Algorithm for Computing the Interleaving Distance between Merge Trees via Path-Preserving Maps

A merge tree is a fundamental topological structure used to capture the sub-level set (and similarly, super-level set) topology in scalar data analysis. The interleaving distance is a theoretically sound, stable metric for comparing merge trees. However, computing this distance exactly is NP-hard. First fixed-parameter tractable (FPT) algorithm for it's exact computation introduces the concept of an $\varepsilon$-good map between two merge trees, where $\varepsilon$ is a candidate value for the interleaving distance. The complexity of their algorithm is $O(2^{2τ}(2τ)^{2τ+2}\cdot n^2\log^3n)$ where $τ$ is the degree-bound parameter and $n$ is the total number of nodes in both the merge trees. Their algorithm exhibits exponential complexity in $τ$, which increases with the increasing value of $\varepsilon$. In the current paper, we propose an improved FPT algorithm for computing the $\varepsilon$-good map between two merge trees. Our algorithm introduces two new parameters, $η_f$ and $η_g$, corresponding to the numbers of leaf nodes in the merge trees $M_f$ and $M_g$, respectively. This parametrization is motivated by the observation that a merge tree can be decomposed into a collection of unique leaf-to-root paths. The proposed algorithm achieves a complexity of $O\!\left(n^2\log n+η_g^{η_f}(η_f+η_g)\, n \log n \right)$. To obtain this reduced complexity, we assume that number of possible $\varepsilon$-good maps from $M_f$ to $M_g$ does not exceed that from $M_g$ to $M_f$. Notably, the parameters $η_f$ and $η_g$ are independent of the choice of $\varepsilon$. Compared to their algorithm, our approach substantially reduces the search space for computing an optimal $\varepsilon$-good map. We also provide a formal proof of correctness for the proposed algorithm.

cs.CG↗
arXiv2026

An Algorithm for Fast and Correct Computation of Reeb Spaces for PL Bivariate Fields

The Reeb space is a fundamental data structure in computational topology that represents the fiber topology of a multi-field (or multiple scalar fields), extending the level set topology of a scalar field. Efficient algorithms have been designed for computing Reeb graphs, however, computing correct Reeb spaces for PL bivariate fields, is a challenging open problem. There are only a few implementable algorithms in the literature for computing Reeb space or its approximation, via range quantization or by computing a Jacobi fiber surface, which are computationally expensive or have correctness issues, i.e., the computed Reeb space may not be topologically equivalent or homeomorphic to the actual Reeb space. In the current paper, we propose a novel algorithm for fast and correct computation of the Reeb space corresponding to a generic PL bivariate field defined on a triangulation $\mathbb{M}$ of a $3$-manifold without boundary, leveraging the fast algorithms for computing Reeb graphs in the literature. Our algorithm is based on the computation of a Multi-Dimensional Reeb Graph (MDRG) which is first proved to be homeomorphic with the Reeb space. For the correct computation of the MDRG, we compute the Jacobi set of the PL bivariate field and its projection into the Reeb space, called the Jacobi structure. Finally, the correct Reeb space is obtained by computing a net-like structure embedded in the Reeb space and then computing its $2$-sheets in the net-like structure. The time complexity of our algorithm is $\mathcal{O}(n^2 + n\, c_{int}\, \log n + nc_L^2)$, where $n$ is the total number of simplices in $\mathbb{M}$, $c_{int}$ is the number of intersection points of the projections of the non-adjacent Jacobi set edges on the range of the bivariate field and $c_L$ is the upper bound on the number of simplices in the link of an edge of $\mathbb{M}$.

cs.CG↗
arXiv2025

Cobordism of algebraic knots defined by Brieskorn polynomials, II

In our previous paper, we obtained several results concerning cobordisms of algebraic knots associated with Brieskorn polynomials: for example, under certain conditions, we showed that the exponents are cobordism invariants. In this paper, we further obtain new results concerning the Fox--Milnor type relations, decomposition of the algebraic cobordism class of an algebraic knot associated with a Brieskorn polynomial that has a null-cobordant factor over the field of rational numbers, and cyclic suspensions of knots. As a corollary, we show that a spherical algebraic knot associated with a Brieskorn polynomial has infinite order in the knot cobordism group.

math.GT↗
arXiv2025

Simplifying generic smooth maps to the 2-sphere and to the plane

We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ to the $2$--sphere $S^2$ and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable map with at most one cusp point and with only folds of the middle absolute index. Furthermore, if $n$ is even, such a $C^\infty$ stable map can be so constructed that the restriction to the singular point set is a topological embedding. As a corollary, we show that for $n \geq 2$ even, there always exists a $C^\infty$ stable map $M \to \mathbf{R}^2$ with at most one cusp point such that the restriction to the singular point set is a topological embedding. As another corollary, we give a new proof to the existence of an open book structure on odd dimensional manifolds which extends a given one on the boundary, originally due to Quinn. Finally, using the open book structure thus constructed, we show that $k$--connected $n$--dimensional manifolds always admit a fold map into $\mathbf{R}^2$ without folds of absolute indices $i$ with $1 \leq i \leq k$, for $n \geq 7$ odd and $1 \leq k \leq (n-5)/2$.

math.GT↗
arXiv2024

Reeb Complements for Exploring Inclusions Between Isosurfaces From Two Scalar Fields

This article proposes to integrate two Reeb graphs with the information of their isosurfaces' inclusion relation. As computing power evolves, there arise numerical data that have small-scale physics inside larger ones -- for example, small clouds in a simulation can be contained inside an atmospheric layer, which is further contained in an enormous hurricane. Extracting such inclusions between isosurfaces is a challenge for isosurfacing: the user would have to explore the vast combinations of isosurfaces $(f_1^{-1}(l_1), f_2^{-1}(l_2))$ from scalar fields $f_i: M(n) \to \mathbb{R}$, $i = 1, 2$, where $M$ is an $n$-dimensional domain manifold and $f_i$ are physical quantities, to find inclusion of one isosurface within another. For this, we propose the \textit{Reeb complement}, a topological space that integrates two Reeb graphs with the inclusion relation. The Reeb complement has a natural partition that classifies equivalent containment of isosurfaces. This is a handy characteristic that lets the Reeb complement serve as an overview of the inclusion relationship in the data. We also propose level-of-detail control of the inclusions through simplification of the Reeb complement. We demonstrate that the relationship of two independent scalar fields can be extracted by taking the product of Reeb graphs (which we call the Reeb product) and by then subtracting the projection of the Reeb space, which opens up a new possibility for feature analysis.

cs.CG↗
arXiv2024

Differentiable maps on links of complex isolated hypersurface singularities

We consider links of complex isolated hypersurface singularities in $\mathbb{C}^{n+1}$ and study differentiable maps defined by restricting holomorphic functions to the links. We give an explicit example in which such a restriction gives a fold map into the plane $\mathbb{C} = \mathbb{R}^2$ whose singular value set consists of concentric circles.

math.GT↗
arXiv2023

Reeb spaces of smooth functions on manifolds II

The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we characterize those smooth functions on closed manifolds whose Reeb spaces have the structure of a finite graph. We also give several explicit examples of smooth functions on closed manifolds such that they themselves or their Reeb spaces have some interesting properties.

math.GT↗
arXiv2023

Special generic maps and Gromoll filtration

A smooth map of a closed $n$-dimensional manifold into $\mathbf{R}^p$ with $1 \leq p \leq n$ is a special generic map if it has only definite folds as its singularities. We show that for $1 \leq p < n$ and $n \geq 6$, a homotopy $n$-sphere admits a special generic map into $\mathbf{R}^p$ with standard properties if and only if its Gromoll filtration is equal to $p$.

math.GT↗
arXiv2021

Round fold maps of n--dimensional manifolds into ${\mathbb{R}}^{n-1}$

We determine those smooth $n$--dimensional closed manifolds with $n \geq 4$ which admit round fold maps into ${\mathbb{R}}^{n-1}$, i.e.\ fold maps whose critical value sets consist of disjoint spheres of dimension $n-2$ isotopic to concentric spheres. We also classify such round fold maps up to $C^{\infty}$ $\mathcal{A}$--equivalence.

math.GT↗
arXiv2021

Round fold maps on $3$--manifolds

We show that a closed orientable 3--dimensional manifold admits a round fold map into the plane, i.e. a fold map whose critical value set consists of disjoint simple closed curves isotopic to concentric circles, if and only if it is a graph manifold, generalizing the characterization for simple stable maps into the plane. Furthermore, we also give a characterization of closed orientable graph manifolds that admit directed round fold maps into the plane, i.e.\ round fold maps such that the number of regular fiber components of a regular value increases toward the central region in the plane.

math.GT↗
arXiv2020

Reeb spaces of smooth functions on manifolds

The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.

math.GT↗
arXiv2018

Elimination of definite fold II

In this paper, we first give a new simple proof to the elimination theorem of definite fold by homotopy for generic smooth maps of manifolds of dimension strictly greater than $2$ into the $2$--sphere or into the real projective plane. Our new proof has the advantage that it is not only constructive, but is also algorithmic: the procedures enable us to construct various explicit examples. We also study simple stable maps of $3$--manifolds into the $2$--sphere without definite fold. Furthermore, we prove the non-existence of singular Legendre fibrations on $3$--manifolds, answering negatively to a question posed in our previous paper.

math.GT↗
arXiv2018

Linking between singular locus and regular fibers

Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $\mathbb{R}^2$.

math.GT↗
arXiv2018

An algebraic description of screw dislocations in SC and BCC crystal lattices

We give an algebraic description of screw dislocations in a crystal, especially simple cubic (SC) and body centered cubic (BCC) crystals, using free abelian groups and fibering structures. We also show that the strain energy of a screw dislocation based on the spring model is expressed by the Epstein-Hurwitz zeta function approximately.

math-ph↗
arXiv2017

Simplified broken Lefschetz fibrations and trisections of 4-manifolds

Shapes of four dimensional spaces can be studied effectively via maps to standard surfaces. We explain, and illustrate by quintessential examples, how to simplify such generic maps on 4-manifolds topologically, in order to derive simple decompositions into much better understood manifold pieces. Our methods not only allow us to produce various interesting families of examples, but also to establish a correspondence between simplified broken Lefschetz fibrations and simplified trisections of closed, oriented 4-manifolds.

math.GT↗
arXiv2017

Simplifying indefinite fibrations on 4-manifolds

We present explicit algorithms for simplifying the topology of indefinite fibrations on 4-manifolds, which include broken Lefschetz fibrations and indefinite Morse 2-functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1-parameter families. In particular, given an arbitrary broken Lefschetz fibration, we show how to turn it to one with directed and embedded round (indefinite fold) image, and to one with all the fibers and the round locus connected. We also show how to realize any given null-homologous 1-dimensional submanifold with prescribed local models for its components as the round locus of such a broken Lefschetz fibration. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2-functions on general 4-manifolds, and a theorem of Auroux-Donaldson-Katzarkov on the existence of broken Lefschetz pencils with directed embedded round image on near-symplectic 4-manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay-Kirby trisections of 4-manifolds, and show the existence of simplified trisections on all 4-manifolds. Building on this correspondence, we provide several new constructions of trisections, including infinite families of genus-3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4-manifolds in the homeomorphism classes of complex rational surfaces.

math.GT↗
arXiv2015

Multivariate Topology Simplification

Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multi-field data, topological analysis requires simultaneous advances both mathematically and computationally. We propose a robust multivariate topology simplification method based on ``lip''-pruning from the Reeb Space. Mathematically, we show that the projection of the Jacobi Set of multivariate data into the Reeb Space produces a Jacobi Structure that separates the Reeb Space into simple components. We also show that the dual graph of these components gives rise to a Reeb Skeleton that has properties similar to the scalar contour tree and Reeb Graph, for topologically simple domains. We then introduce a range measure to give a scaling-invariant total ordering of the components or features that can be used for simplification. Computationally, we show how to compute Jacobi Structure, Reeb Skeleton, Range and Geometric Measures in the Joint Contour Net (an approximation of the Reeb Space) and that these can be used for visualisation similar to the contour tree or Reeb Graph.

cs.CG↗