In this paper we prove that any triangulation of a 2-dimensional sphere with a strict 4-coloring on its vertices can seen as the boundary of a triangulation of a 3-dimensional disk with the same vertices and preserving the 4-coloring.
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11 results for “4-colorings”
The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric …
We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the eighties by S. Lins for closed 3-manifolds, which …
Study of unoriented SL(4) foams in 3-manifolds.
problem Understanding unoriented SL(4) foams in 3-manifolds.
method Combinatorial evaluation and state space study of unoriented SL(4) foams.
result State space of any web is free of rank given by the number of its 4-colorings over a localized ground ring.
New proof of a 111-year-old result using gauge theory.
problem Proving configurations in the list of unavoidable configurations are reducible.
method Filtered 3- and 4-color homology, gauge theory, state-reducibility.
result Birkhoff diamond is reducible using gauge theory.
A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of th…
-Holonomy on Coloring Complexes of with Applications to the Poincaré Conjecture and -Color Theoremmath.GT
A natural class of coloring complexes on closed manifold is investigated that gives a holonomy map $\mbox{Hol}_X: π_1(M) \to S_{n+1}$. By a -multilayer complex construction the holonomy map may be defined to any finite permutation group $\mbox{Hol}_X: π_1(M) \to S_{n+k}$, . Under isotopy of and su…
The G-degree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models. The present paper studies the properties of the G-degree for the large class of graphs representing singular manifolds (including closed PL manifolds). In particular, the complete topological classification up to G-deg…
This paper is a collection of thoughts and observations, being partly a review and partly a report of current research, on recent work in various aspects of Grünbaum colorings, their existence and usage. In particular, one of the most striking significances of Grünbaum's Conjecture in the 2-dimensional case is its equi…
The paper shows links can be colored with fewer colors than previously thought.
problem Coloring links using the symmetric group of degree three.
method Analyzing the number of colors for link colorings by .
result 2-bridge links with 5 colors can be colored with only 4 colors.
Stochastic Multi-level Composition Optimization Algorithms with Level-Independent Convergence Ratesmath.OC
New algorithms solve complex multi-level optimization problems with improved efficiency.
problem Smooth stochastic multi-level composition optimization problems.
method Two algorithms using moving-average and linearized stochastic estimates.
result Achieved sample complexities of O(1/ε^4) and O(1/ε^6).