2012/11/09 by Sóstenes Lins, Ricardo Machado, Lins, Sóstenes +2
Computer Science · Mathematics · #57M15 (Secondary) #57M25 #57Q15 (Primary) 57M27 #Advanced Combinatorial Mathematics #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Topological and Geometric Data Analysis #math.GT #msc:57M15 #msc:57M25 #msc:57M27 #msc:57Q15
paper · pdf · doi:10.48550/arxiv.1212.0827
This is a major revision of part 3 with 25 pages, 29 figures of a 3-part article
openalex publication_date 2012/11/09 · arxiv created 2013/02/23 · arxiv updated 2013/02/26 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28
In this final part of a 3-part paper we introduce the pair of "wings" of the abstract PL-colored complexes Hm^⋆, described in the second paper. The wings, via a weight enhanced Tutte's barycentric embedding of a planar map, produce the unexpected reformutation of a 3-dimensionl problem into a 2-dimensional one. The total number of edges in each one of the pair of final wings is less than 8n-5. Tutte's method is applied O(n) times to each one of the 2 wings in the final pair to assure rectilinearity of the embeddings of the planar maps, which include the final wings. A cone construction over the final wings provides a PL-complex H1^\diamond, which contain the set of 0-simplices \a1, a2,...,af\ ∪ \b1, b2,...,bg\ (as defined in the second part of the article) properly fixed in ℝ3. The other 0-simplices are obtained by bisections of segments linking previously defined points. This implies that Hn is PL-embedded into ℝ3. We then conclude the surgery description of the 3-manifold induced by the gem with its resolution by defining some disjoint cylinders contained in Hn^⋆, directly from the hinges (dual of the twistors of the resolution), in a 1-1 correspondence. The medial curves of the cylinders define the link we seek. The framing of a medial curve is the linking number of the boundary components of the corresponding cylinder. The analysis of the whole proccess shows that the memory and time requirement to complete the algorithm is O(n2). Data for the Weber-Seifert 3-manifold, which answers Jeffrey Weeks's question is given in the appendix. It consists of a link with 142 crossings but it admits simplifications.