Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions

Abstract

We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obtained from the Laplacian block matrices by means of Gauss partial diagonalization procedure described explicitly by W. Neumann. The linking matrix is an important topological invariant of a graph manifold which is possible to interpret as a matrix of coupling constants of gauge interaction in Kaluza-Klein approach, where 3-dimensional graph manifold plays the role of internal space in topological 7-dimensional BF theory. The Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices.

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

“…This algorithm gives us the possibility to calculate (with a few operations) the complete set of integer Euler numbers (the principal topological invariant) of sophisticate graph manifolds, which we used to simulate the coupling constant hierarchy of the fundamental interactions acting in our universe [13]. Automatically we can explicitly compute the integer Laplacian block matrix associated with any tree plumbing graph [5]. This matrix coincides up to sign with the integer linking matrix of the graph manifold corresponding to the plumbing graph [14].…”

“…This matrix coincides up to sign with the integer linking matrix of the graph manifold corresponding to the plumbing graph [14]. The rational linking matrix (describing the coupling constants hierarchy) can be obtained from the Laplacian one by means of Gauss-Neumann partial diagonalization procedure described explicitly in [5] [17]. It makes sense to emphasize that the rank of the integer Laplacian block matrix, corresponding to the realistic model coupling constants hierarchy is of order 10 20 .…”

“…In this section, we review the main concepts of codifying the topology of plumbing graph manifolds by means of continued fraction expansion (for more detales see [1] [2] [5]). For given coprime integers p and q, the Hirzebruch-Jung modification of euclidean algorithm …”

“…With the aim of obtain the coupling constant hierarchy we consider tree plumbing graphs p Γ of more general type [5] [13], whose basic structure blocks are Seifert fibered Brieskorn homology spheres [7]. Let 1 2 3 , , a a a be pairwise relatively prime positive numbers, the Brieskorn homology sphere (Bh-sphere) is defined as the link of Brieskorn singularity ( ) ( ) { } 5 3 1 2 1 2 3 1 2 3 : , , : , , a a a Σ is given in Figure 2 [1].…”

“…First of all, HJ-continued fractions arise naturally in the minimal resolution of cyclic quotient (that is, Hirzebruch-Jung) surface singularities of the type p   , which is also known as HJresolution [1] [2]. This type of resolution gives rise to a smooth 4-dimensional manifold ( ) , X p q called HJspace [3], which is homeomorphic to a plumbing graph manifold ( ) p P Γ described in [4] [5]. The boundary of HJ-space…”