Algorithm for Fast Calculation of Hirzebruch-Jung Continued Fraction Expansions to Coding of Graph Manifolds

• 作者: Alfonso M. Hernández Magdaleno Fernando I. Becerra López Vladimir N. Efremov
• 来源期刊: 应用数学(英文)
• 学科: 数学
• 关键词: Hirzebruch-Jung Continued Fraction Fast Expansion ALGORITHM GRAPH MANIFOLDS
• 年，卷（期）: 2015,（10）
• 页码范围: 1676-1684
• 页数: 9页
• 分类号: O1

摘要

We present a new algorithm for the fast expansion of rational numbers into continued fractions. This algorithm permits to compute the complete set of integer Euler numbers of the sophisticate tree graph manifolds, which we used to simulate the coupling constant hierarchy for the universe with five fundamental interactions. Moreover, we can explicitly compute the integer Laplacian block matrix associated with any tree plumbing graph. This matrix coincides up to sign with the integer linking matrix (the main topological invariant) of the graph manifold corresponding to the plumbing graph. The need for a special algorithm appeared during computations of these topological invariants of complicated graph manifolds since there emerged a set of special rational numbers (fractions) with huge numerators and denominators;for these rational numbers, the ordinary methods of expansion in continued fraction became unusable.