arXiv subjects
Publications and source records attributed to Ricardo Machado.
In this article we focus on the solutions of a congruence equation: $\binom{2n}{n}\equiv 2^n \bmod n$. Using the main result of this article and the SageMath software, we improve largely the number of known solutions. Furthermore, we prove that some famous numbers like even perfect numbers and Wieferich primes are connected to solutions of this equation.
We give a complete list of square-free Cremona maps with at most six variables, up to equivalence classes. We also build an algorithm to count monomial square-free Cremona transformations. Using this algorithm, we obtain a complete list of monomial square-free Cremona transformations in seven variables.
In this final part of a 3-part paper we introduce the pair of "wings" of the abstract PL-colored complexes $\mathcal{H}_{m}^\star$, 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 $\mathcal{H}_1^\diamond$, which contain the set of 0-simplices $\{a_1, a_2,...,a_f\} \cup \{b_1, b_2,...,b_g\}$ (as defined in the second part of the article) properly fixed in $\mathbb{R}^3$. The other 0-simplices are obtained by bisections of segments linking previously defined points. This implies that $\mathcal{H}_n$ is PL-embedded into $\mathbb{R}^3$. We then conclude the surgery description of the 3-manifold induced by the gem with its resolution by defining some disjoint cylinders contained in $\mathcal{H}_{n}^\star$, 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(n^2). 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.
Given an special type of triangulation $T$ for an oriented closed 3-manifold $M^3$ we produce a framed link in $S^3$ which induces the same $M^3$ by an algorithm of complexity $O(n^2)$ where $n$ is the number of tetrahedra in $T$ . The special class is formed by the duals of the {\em solvable gems}. These are in practice computationaly easy to obtain from any triangulation for $M^3$. The conjecture that each closed oriented 3-manifold is induced by a solvable gem has been verified in an exhaustible way for manifolds induced by gems with few vertices. Our algorithm produces framed link presentations for well known 3-manifolds which hitherto did not one explicitly known. A consequence of this work is that the 3-manifold invariants which are presently only computed from surgery presentations (like the Witten-Reshetkhin-Turaev invariant) become computable also from triangulations. This seems to be a new and useful result. Our exposition is partitioned into 3 articles. This first article provides our motivation, some history on presentation of 3-manifolds and recall facts about gems which we need.
This is part 2 of a 3-part article where we provide an $O(n^2)$-algorithm to produce a surgery presentation of a 3-manifold induced by a gem with a resolution. In this part we produce a sequence of colored simplicial 2-complexes which are inverses and dual to the sequence of gems produced in the first part. The refinements of the DPL-faces that keep appearing are idempotent: the second refinement of a DPL-face is isomorphic to its first refinement. This fact inhibits exponentiability.