Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0710.0080

Equivalent metrics and compactifications

Young Deuk Kim
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let (X,d) be a metric space and m\in X. Suppose that ϕ:X\times X\to\mathbold{R} is a nonnegative symmetric function. We define a metric d^{ϕ,m} on X which is equivalent to d. If d^{ϕ,m} is totally bounded, its completion is a compactification of (X,d). As examples, we construct two compactifications of (\mathhbold{R}^s,d_E), where d_E is the Euclidean metric and s\geq 2.

Explore related subjects

math.GTmath.GN

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Young Deuk Kim. 2007-09-29. Equivalent metrics and compactifications. https://arxiv.org/abs/0710.0080

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Causality Detection via Symplectic Quandles

We study whether symplectic quandle colorings can reveal causal structure encoded by "sky links", the links formed by the spheres of all light rays through two events in the space of light rays of a $(2+1)$-dimensional globally hyperbolic spacetime. The testbed is the connected sum of two Hopf links $H$ (causally unrelated events) and the infinite family of Allen-Swenberg links $L_n$, which the Alexander-Conway polynomial cannot tell apart from $H$. We use the enhanced quandle counting polynomial, in which every coloring is weighted by the size of the subquandle generated by its colors. For the 16-element symplectic quandle $T=(\mathbb{Z}/4)^2$ we prove that $|\mathrm{Hom}(Q(L_n),T)| = 640 + 96\cdot 16^n$ for every $n\ge 1$, while $|\mathrm{Hom}(Q(H),T)| = 736$, and we determine the enhanced polynomial of every $L_n$. Hence already the plain quandle counting invariant distinguishes $H$ from every Allen-Swenberg link, and the Allen-Swenberg links from each other. Analogous closed formulas hold for $(\mathbb{Z}/2)^4$ and for an 8-element degenerate symplectic quandle $(\mathbb{Z}/2)^3$ ($176+48\cdot 16^n$ versus $224$), and the quandle $(\mathbb{Z}_7)^2$ also separates $H$ from the whole family. Over $(\mathbb{Z}_p)^2$ the enhanced polynomial carries at most one number beyond the counting invariant, and whether it separates $L_1$ from $H$ depends on $p$: it does for $p=7,11,17,19,23,31$ and does not for $p=2,3,5,13$. The proofs combine an exact transfer-matrix description of the family with certified finite computations.

math.GT↗
arXiv2026

Cubical structures and the large-scale geometry of graph braid groups

We study the large-scale geometry of graph braid groups through the cubical structure of their unordered discrete configuration spaces, focusing on quasi-isometry to right-angled Artin groups (RAAGs). We first give a complete classification of graph braid groups quasi-isometric to free groups. For the $2$-braid group on a graph $Γ$, we study the union $UP_2(Γ)$ of maximal product subcomplexes of the associated unordered discrete configuration space. We introduce a hierarchy recording geometric and algebraic properties of this inclusion and identify conditions under which the quasi-isometry type of its fundamental group is determined by that of the ambient braid group. For normal bunches of grapes, a class of graphs obtained by attaching cycles to trees, this fundamental group is a one-ended free factor, and the quasi-isometry classification of the braid groups reduces to that of these factors. Using the combinatorics of the underlying trees and the associated intersection complexes, we obtain a graph-theoretic sufficient condition and new obstructions for quasi-isometry to RAAGs, yielding infinite families of non-hyperbolic examples and nonexamples. We also construct infinitely many graph $2$-braid groups hyperbolic relative to a thick proper subgroup not isomorphic to any braid group with at most two particles on a subgraph of the underlying graph.

math.GT↗
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.

math.GT↗