Tootfinder

Minors of non-hamiltonian polyhedra and the Herschel family
On-Hei Solomon Lo, Kenta Ozeki
https://arxiv.org/abs/2506.20086 https://a…

Minors of non-hamiltonian polyhedra and the Herschel family
We show that every non-hamiltonian polyhedron contains the Herschel graph as a minor. As an application, we provide a characterization of non-hamiltonian polyhedra without $K_{2,6}$ minors, thereby resolving a conjecture by Ellingham, Marshall, Ozeki, Royle, and Tsuchiya.

Parallel Polyhedral Projection Method for the Convex Feasibility Problem
Pablo Barros, Roger Behling, Vincent Guigues
https://arxiv.org/abs/2506.15895 http…

Parallel Polyhedral Projection Method for the Convex Feasibility Problem
In this paper, we introduce and study the Parallel Polyhedral Projection Method (3PM) and the Approximate Parallel Polyhedral Projection Method (A3PM) for finding a point in the intersection of finitely many closed convex sets. Each iteration has two phases: parallel projections onto the target sets (exact in 3PM, approximate in A3PM), followed by an exact or approximate projection onto a polyhedron defined by supporting half-spaces. These strategies appear novel, as existing methods largely fo…

Steffen's flexible polyhedron is embedded. A proof via symbolic computations
Victor Alexandrov, Evgenii Volokitin
https://arxiv.org/abs/2508.02392 https://

Steffen's flexible polyhedron is embedded. A proof via symbolic computations
A polyhedron is flexible if it can be continuously deformed preserving the shape and dimensions of every its face. In the late 1970's Klaus Steffen constructed a sphere-homeomorphic embedded flexible polyhedron with triangular faces and with 9 vertices only, which is well-known in the theory of flexible polyhedra. At about the same time, a hypothesis was formulated that the Steffen polyhedron has the least possible number of vertices among all embedded flexible polyhedra without boundary. A cou…

Can we "seamlessly" divide a polygon?
Byungchang So
https://arxiv.org/abs/2506.11742 https://arxiv.org/pdf/2506.11742

Can we "seamlessly" divide a polygon?
While the contents of Euclid's Elements are well-known these days, some characters of the original text have been overlooked due to interpretation by modern mathematical languages. The lens of modern mathematics once anachronistically misled researchers of the history of mathematics, and this shows that the classic text itself contains ideas that have not been completely accurately translated. This article concentrates on the division of geometric figures(line, polygon, polyhedron, etc.). In mo…

Flows on Gentle Algebras
Jonah Berggren
https://arxiv.org/abs/2507.12688 https://arxiv.org/pdf/2507.12688

Flows on Gentle Algebras
The space of unit flows on a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. Amply framed DAGs and their triangulated flow polytopes have recently been connected with the representation theory of certain gentle algebras. We expand on this connection by defining a flow on the fringed quiver of an arbitrary gentle algebra. We call the space of unit flows its turbulence polyhedron. We show that support tau-tilting modules of a gentle…

Toric reduction of singularities for Newton non-degenerate $p$-forms
Bilal Balo
https://arxiv.org/abs/2507.05430 https://arxiv.org/pd…

Toric reduction of singularities for Newton non-degenerate $p$-forms
We study a class of holomorphic $p$-forms satisfying non-degeneracy conditions expressed through their Newton polyhedron. We show that a regular refinement of their dual fan defines a toric reduction of singularities to a log-smooth $p$-form.

Decremental Greedy Polygons and Polyhedra Without Sharp Angles
David Eppstein
https://arxiv.org/abs/2507.04538 https://arxiv.org/pdf/…

Decremental Greedy Polygons and Polyhedra Without Sharp Angles
We show that the max-min-angle polygon in a planar point set can be found in time $O(n\log n)$ and a max-min-solid-angle convex polyhedron in a three-dimensional point set can be found in time $O(n^2)$. We also study the maxmin-angle polygonal curve in 3d, which we show to be $\mathsf{NP}$-hard to find if repetitions are forbidden but can be found in near-cubic time if repeated vertices or line segments are allowed, by reducing the problem to finding a bottleneck cycle in a graph. We formalize …

Knot complements decomposing into prisms
Jason Deblois, Arshia Gharagozlou, Neil R Hoffman
https://arxiv.org/abs/2507.01263 https://a…

Knot complements decomposing into prisms
We describe four hyperbolic knot complements in $\mathbb{S}^3$, each of which covers a prism orbifold: the quotient of $\mathbb{H}^3$ by the action of a discrete group generated by reflections in the faces of a polyhedron that has the combinatorial type of a triangular prism. The prism orbifolds are rigid-cusped and contain compact, totally geodesic hyperbolic triangle sub-orbifolds; as a result, the knot complements covering them have hidden symmetries and contain closed, embedded, totally geo…

Supersymmetric Schur polynomials have saturated Newton polytopes
Dang Tuan Hiep, Khai-Hoan Nguyen-Dang
https://arxiv.org/abs/2507.22528 https://arxiv.org/p…

Supersymmetric Schur polynomials have saturated Newton polytopes
We prove that every supersymmetric Schur polynomial has a saturated Newton polytope (SNP). Our approach begins with a tableau-theoretic description of the suport, which we encode as a polyhedron with a totally unimodular constraint matrix. The integrality of this polyhedron follows from the Hoffman-Kruskal criterion, thereby establishing the SNP property. To our knowledge, this is the first application of total unimodularity to the SNP problem.

Potential series expansion method applied in Analytical Modeling of Gravitational field of Irregularly Shaped Celestial Bodies
Marcelo Lisboa Mota, Safwan Aljbaae, Antonio F. B. A. Prado
https://arxiv.org/abs/2507.19433

Potential series expansion method applied in Analytical Modeling of Gravitational field of Irregularly Shaped Celestial Bodies
This study aims to establish an analytical model that reproduces the gravitational field around non-spherical bodies with constant density. Due to the non-spherical geometry of such bodies, their gravitational potential is disturbed relative to a central field. By considering the body as a polyhedron and decomposing it into tetrahedral elements, we use the Series Potential Expansion Method (PSEM) to approximate the total potential by summing the potentials of each tetrahedron. While this model …

Weighted blow-up desingularization of plane curves
Dan Abramovich, Ming Hao Quek, Bernd Schober
https://arxiv.org/abs/2507.01232 https://

Weighted blow-up desingularization of plane curves
Given a singular hypersurface in a regular 2-dimensional scheme essentially of finite type over a field, we construct an embedded resolution of singularities by tame Artin stacks. This applies in particular to plane curves. The key tool is Hironaka's characteristic polyhedron from which we determine a suitable center for a stack-theoretic weighted blow-up. In particular, we show that the order strictly decreases at every point lying above the center. In contrast to earlier work by the authors, …

Shellings of Tropical Hypersurfaces
George Balla, Michael Joswig, Lena Weis
https://arxiv.org/abs/2506.07241 https://arxiv.org/pdf/25…

Shellings of Tropical Hypersurfaces
The shellability of the boundary complex of an unbounded polyhedron is investigated. To this end, it is necessary to pass to a suitable compactification, e.g., by one point. This observation can be exploited to prove that any tropical hypersurface is shellable. Under the hood there is a subtle interplay between the duality of polyhedral complexes and their shellability. Translated into discrete Morse theory, that interplay entails that the tight span of an arbitrary regular subdivision is colla…

1-Uryson width and covers
Hannah Alpert, Arka Banerjee, Panos Papasoglu
https://arxiv.org/abs/2505.21126 https://arxiv.org/pdf/2505.2…

1-Uryson width and covers
We investigate the following question: Do there exist Riemannian polyhedra $X$ such that the 1-Uryson width of their universal covers $\mathrm{UW}_1(\widetilde{X})$ is bounded but $\mathrm{UW}_1(X)$ is arbitrarily large? We rule out two specific cases: when $π_1(X)$ is virtually cyclic and when $X$ is a Riemannian surface. More specifically, we show that if $X$ is a compact polyhedron with a virtually cyclic fundamental group, then its 1-Uryson width is bounded by the 1-Uryson width of its uni…