Related papers: Cut Locus Realizations on Convex Polyhedra

An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the…

We prove that if a knot $K$ has a particular type of diagram then all non-trivial surgeries on $K$ contain a coorientable taut foliation. Knots admitting such diagrams include many two-bridge knots, many pretzel knots, many Montesinos knots…

We show that for every injective continuous map f: S^2 --> R^3 there are four distinct points in the image of f such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four…

Using the theory of plugs and the self-insertion construction due to the second author, we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets…

Given fixed distinct points $A, B, C, D$, we examine properties of the locus of points $X$ for which $(XA, XC)$, $(XB, XD)$ are isogonal. This locus is a cubic curve circumscribing $ABCD$. We characterize all possible such cubics $\mathcal…

In the present paper, we propose a novel generalization of the celebrated MMP algorithm in order to find the wavefront propagation and the cut-locus on a convex polyhedron with an emphasis on actual implementation for instantaneous…

We present two algorithms for unfolding the surface of any polyhedron, all of whose faces are triangles, to a nonoverlapping, connected planar layout. The surface is cut only along polyhedron edges. The layout is connected, but it may have…

Let $P$ be a finite poset. We will show that for any reasonable $P$-persistent object $X$ in the category of finite topological spaces, there is a $P-$ weighted graph, whose clique complex has the same $P$-persistent homology as $X$.

We show that Cutting Planes (CP) proofs are hard to find: Given an unsatisfiable formula $F$, 1) It is NP-hard to find a CP refutation of $F$ in time polynomial in the length of the shortest such refutation; and 2)unless Gap-Hitting-Set…

By Bartle-Graves theorem every surjective map between C*-algebras has a continuous section, and Loring proved that that there exists a continuous section of norm arbitrary close to 1. Here we prove that there exists a continuous section of…

We study the stable pairs theory of local curves in 3-folds with descendent insertions. The rationality of the partition function of descendent invariants is established for the full local curve geometry (equivariant with respect to the…

The straight skeleton of a polygon is the geometric graph obtained by tracing the vertices during a mitered offsetting process. It is known that the straight skeleton of a simple polygon is a tree, and one can naturally derive directions on…

We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types.…

We prove that every tree of maximum degree $\Delta$ with $\ell$ leaves contains paths between leaves of at least $\log_{\Delta-1}((\Delta-2)\ell)$ distinct lengths. This settles in a strong form a conjecture of Narins, Pokrovskiy and…

We attempt to shed new light on the notion of 'tree-like' metric spaces by focusing on an approach that does not use the four-point condition. Our key question is: Given metric space $M$ on $n$ points, when does a fully labelled…

Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. We first consider…

We present an explicit closed-form formula for the vertices of the classical cut polytope $\operatorname{CUT}(n)$, defined as the convex hull of cut vectors of the complete graph $K_n$. Our derivation proceeds via a related polytope,…

Let $T$ be a tree, a vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. For two distinct vertices $u,v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v.$ For a…

I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve.…

Gomory and Hu proved that if $ G $ is a finite graph with non-negative weights on its edges, then there exists a tree $ T $ (called now Gomory-Hu tree) on $ V(G) $ such that for all $ u\neq v\in V(G) $ there is an $ e\in E(T) $ such that…