2007/01/20 by Sóstenes Lins, Sostenes Lins, Lins, Sostenes
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.GT
paper · pdf · doi:10.48550/arxiv.math/0701578
arxiv created 2007/01/20 · openalex publication_date 2007/01/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
From a pseudo-triangulation with n tetrahedra T of an arbitrary closed orientable connected 3-manifold (for short, \em a 3D-space) M3, we present a gem J ', inducing \IS3, with the following characteristics: (a) its number of vertices is O(n); (b) it has a set of p pairwise disjoint couples of vertices \ui,vi\, each named \em a twistor; (c) in the dual (J ')^⋆ of J ' a twistor becomes a pair of tetrahedra with an opposite pair of edges in common, and it is named \em a hinge; (d) in any embedding of (J ')^⋆ ⊂ \IS3, the ε-neighborhood of each hinge is a solid torus; (e) these p solid tori are pairwise disjoint; (f) each twistor contains the precise description on how to perform a specific surgery based in a Denh-Lickorish twist on the solid torus corresponding to it; (g) performing all these p surgeries (at the level of the dual gems) we produce a gem G ' with |G '|=M3; (h) in G ' each such surgery is accomplished by the interchange of a pair of neighbors in each pair of vertices: in particular, |V(G ')=|V(J ')|. This is a new proof, \em based on a linear polynomial algorithm, of the classical Theorem of Wallace (1960) and Lickorish (1962) that every 3D-space has a framed link presentation in \IS3 and opens the way for an algorithmic method to actually obtaining the link by an O(n2)-algorithm. This is the subject of a companion paper soon to be released.