Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions ()
1. Introduction
Graphs can serve as a universal remedy for the codification and classification of topological spaces, matrices, dynamical systems, etc. In this article, we consider the following question: how the topological invariants of manifold corresponding to a tree graph (graph manifold) can be calculated using the method of Gauss-Neumann partial diagonalization of Laplacian matrix defined for the same graph. This calculation can be useful in multidimensional models of Kaluza-Klein type, where coupling constants of gauge interactions are simulated by the rational linking matrices of the internal space [1] . We constructed various models [2] [3] where the role of internal spaces is played by a specific family of 3-dimensional graph manifolds, whose rational linking matrices describe the hierarchy of gauge coupling constants of the real universe. The basic structure blocks of these graph manifolds are Seifert fibered Brieskorn homology spheres, defined as the link of Brieskorn singularity
, with 
pairwise relatively prime positive numbers [4] . Bh-spheres belong to the class of Seifert fibered homology spheres (Sfh-spheres). On each of these manifolds, there exists a Seifert fibration with unnormalized Seifert invariants 
subject to 
, where 
and 
is its rational Euler number, the well known topological invariant of a Bh-sphere. The topological operation known as plumbing [5] is used to past together Bh-spheres according to plumbing diagrams (Figure 4) which can be transformed in plumbing graphs (Figure 3). This type of graphs and their Laplacian block matrices is the main object of attention in this article.
The paper is organized as follows. In section 2 we define the type of tree graphs considered in this paper (plumbing graphs) and Laplacian matrices for these plumbing graphs. We recall also the Gauss-Neumann method of partial diagonalization by means of which we obtain the reduced rational tridiagonal matrix for each plumbing graph. In section 3 we construct graph manifolds codified by the plumbing graphs defined in section 2 and calculate the main topological invariant for these 3-dimensional manifolds, namely rational linking matrix. Then 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-Neumann partial diagonalization procedure. Finally, we conclude formulating our main results and considering an example of their application for the topological field theory.
2. Block Matrix Representation for a Graph 
of Tree Type
We begin from the definition of graph 
as a finite one-dimensional simplicial complex, which does not contain multiple edges and loops, i.e. we consider only the graph of tree type. An integer weight 
is assigned to each vertex of
. Vertices with at least three edges are called nodes. For simplicity we shall use graphs with nodes of minimal valence (n = 3) only (a generalization to 
is obvious). Suppose that the set of nodes 
of the graph 
is non-empty. Considering the graph 
as a one-dimensional simplicial complex, we take the complement
. This complement is the disjoint union of straight line segments which are the maximal chains of
. Figure 1 shows a maximal internal chain 
of length k between two nodes 
and
, with weights 
embedded in a tree graph
. The chain is maximal because it cannot be included in some larger chain. Figure 2 shows a maximal terminal chain 
of length k also with weights
.
In this paper we shall considered only the simplest type of graphs which are called plumbing graphs. An example of plumbing graph 
is given in Figure 3. This type of graphs is used to codify the plumbing graph manifolds [5] which will be constructed in the following section, where it will be clear why weights of plumbing graph are called Euler numbers. In Figure 3 the Euler numbers 
and 
decorate the vertices with valence 1 and 2. The nodes are marked by NI with 
and form a straight line or chain structure. We associate
Figure 1. A maximal internal chain of length k.
Figure 2. A maximal terminal chain of length k.
to each node a weight equal to zero that is connected with using of the unnormalized Seifert invariants for Bh-spheres, which are the block elements for the construction of graph manifolds [6] .
Now let’s define a Laplacian matrix for the plumbing graph 
as follows:
with integer numbers 
corresponding to each vertex
. This is a tridiagonal block matrix containing all the information about the graph
. The I-th fragment of the matrix 
which corresponds to the I-th piece of the graph 
shown in Figure 3 is represented as
Note that 
denotes an integer number 0 corresponding to the node
. Now we pay attention to the tridiagonal submatrices (blocks) of type
and notice that using Gauss-Neumann partial diagonalization [7] the matrix is equivalent to the rational block matrix
where
and
, 
,
. Here we are using the standard definition of continued fraction
Applying the general Gauss-Neumann partial diagonalization method for the matrix 
we obtain a similar result where 
is a rational tridiagonal matrix of rank 
(the number of nodes of the graph
) whose elements on the diagonal are a sum of three terms representing each maximal chain connecting to the node.
(1)
where 
are continued fractions for a terminal chain, and 
for a internal chain. We have used the notation 
to indicate that the order of the numbers on the continued fraction is inverse, i.e.
. So, we can reduce each chain of 
to a rational number
, 
or 
which is represented as a continued fraction, and thus reduce the original block tridiagonal matrix 
to a tridiagonal matrix 
whose size depends just of the number of nodes of
. It is important to note that it is possible to obtain the original matrix 
from the reduced matrix
.
3. Rational Linking Matrix for Graph Manifold 
In this section we will construct a plumbing graph manifold 
codified by the same graph 
as in section 2. Now we see the weight 
as the Euler number of the principal S1-(U(1)-)bundle, corresponding to i-th vertex
. We define the bundle 
associated to each vertex 
as S1-bundle over 
with the Euler number
, which can be pasted together from two trivial bundles over 
as follows [8] [9]
where
Note that the above is a well known description of the lens space
, so the total space of the bundle is
. To perform pasting operation, which is known as plumbing between the S1-bundles, we must use the trivial bundles over annuli 
where A is an annulus or twice punctured sphere
. The manifold 
is pasted together from the manifolds 
as follows [9] : whenever vertices 
and 
are connected by an edge 
in 
we paste a boundary component 
of 
to a boundary component 
of 
by the map
:
so the base and fiber coordinates are exchanged under the plumbing operation. Thus the edge 
corresponds to the torus 
along which the pieces 
and 
pasted together.
For example, the plumbing of the chain shown in Figure 1 gives us the pasting
where 
is a 
-bundle associating with the node
. This chain corresponds to a Seifert fibered thick torus (homeomorphic to
) in graph manifold
. Also, the terminal maximal chain shown in Figure 2 corresponds to a Seifert fibered solid torus (homeomorphic to
) in
. The using of graphs with nodes of valence 
(as in Section 2) corresponds to plumbing of Brieskorn homology spheres (Bh-spheres) [4] .
Now recall that each edge 
of 
relates to the embedded torus 
and the collection of all these tori cuts the graph manifold 
into disjoint union of circle bundles over n times punctured sphere 
In general, the bundles are over compact surfaces of genus g with some boundary components, see [8] [9] . Such a collection of tori 
is called a graph structure on 
by Waldhausen [10] . We want to define the Jaco-Shalen-Johannson (JSJ) graph structure 
of the Waldhausen graph structure and to specify the corresponding JSJ-decomposition of graph manifold 
on the set of Seifert fibered pieces
. Let us denote 
the set of maximal chains in the graph
. This set can be written as a disjoint union 
where 
denote the set of interior chains and 
is the set of terminal chains. The edges of 
contained in a chain 
correspond to a set of parallel tori in
. Choose one torus 
among them and define
(2)
This set of tori performs the well known JSJ-decomposition of the graph manifold 
[11] .
By construction, each piece 
(denoted as 
for brevity) of JSJ-decomposition that corresponds to the node 
contains a unique piece 
(which we shall denote as
) of Waldhausen decomposition associated with the same node
. One can extend in a unique way up to isotopy the natural Seifert structure without exceptional fibers on 
to a Seifert fibration on 
with exceptional fibers. Thus in these terms the JSJ-decomposition of the manifold 
is defined completely by 
where R is the number of nodes in 
and the bar over M means the closure of the piece
.
Note that there exists an uncertainty in the choice of the torus 
for each internal chain which appear in the JSJ-structure (2). We can remove this uncertainty in following way. Let us perform the maximal extension of the natural Seifert fibration from each 
and denote the obtained Seifert fibered piece of 
by
. It is clear that 
if and only if there exists a chain 
joining the nodes 
and
. If we start with plumbing of R Bh-spheres
, the resulting graph three-manifold will be integer homology sphere [4] [9] (
-homology sphere), which in general case does not have the global Seifert fibration. But we can construct the JSJ-covering
, such as each 
is a Seifert fibered space and it is maximal in the sense described above.
Suppose that we perform the plumbing operation according to the plumbing diagram
, shown on the Figure 4. Thus our plumbing diagrams will always have the pairwise coprime weights around each node and correspond to 
-homology spheres [5] .
We construct the plumbing graph 
for a 
-homology sphere, following [5] [8] [9] (as a result we shall obtain a graph of type shown in Figure 3). First of all we calculate the characteristics of maximal chains. For terminal chains the integer Euler numbers 
are defined by the continued fraction:
Figure 4. A plumbing (splicing) diagram Δp.
(3)
where
, 
are the Seifert invariants, numerated in the following way
(4)
For internal chains the integer Euler numbers 
are defined by
(5)
where the Seifert (orbital) invariants
, 
characterize the thick tori
, which are created by the plumbing operations performed between the nodes 
and
, see [6] [9] . These invariants identify also the extra lens spaces 
which arise in four-dimensional plumbed V-cobordism (corresponding to the graph
) with
for the ordering fixed by the plumbing diagram in Figure 4. From this representation of the plumbing graph it is clear that for 
the set 
of JSJ-covering 
has the form
(6)
where 
is a Seifert fibered solid torus with Seifert invariants 
and
(7)
For the cases 
and 
the formulas are different from (6):
(8)
(9)
Moreover 
Here the symbol 
indicates that 
and 
are homeomorphic, but their Seifert structures are characterized by different integer Euler numbers defined by (5) and (7) respectively. Thereby the thick torus between the nodes NI and 
has two Seifert fibrations: the first is the extension of the natural Seifert fibration defined on the piece 
and the second one is obtained as extension from the piece
. These Seifert fibrations are connected by the matrix [6] [9] :
in the following sense. Recall that edges of 
contained in a chain 
(between the nodes NI and
) correspond to the set parallel tori in
. On any of these tori there exist two bases formed by the section lines and the fibers pertain to the Seifert fibrations extended from 
and
, which we denote as the pair of columns
(10)
Subindices 2 and 3 manifest that 
and 
are plumbed together along the singular fibers with Seifert invariants 
and 
(see Figure 4). Then the transformation between these section-fiber bases is described by
where 
is defined from
.
Now we introduce the one-form bases 
and
, duals to the bases (10) in the following sense:
where the integrals are calculated over any such section line or fiber as, for example, in [12] . Thus we obtain the corresponding transformations between the the dual one-forms:
(11)
We suppose that the forms 
and 
are dual with respect to the bilinear pairing defined as
(12)
Also we shall used the integrals
which define the linking (intersection) numbers of the fiber structures 
and 
defined on thick torus
. We can obtain the rational linking matrix for 
by means of multiplication of the Equations (11) by 
and 
and integration over
. Applying the duality conditions (12) we obtain:
(13)
The rational numbers 
and 
are also known as Chern classes of the line V-bundles associated with the Seifert fibrations with the U(1)-invariant connection forms 
and 
on the lens spaces 
and 
respectively [12] .
Now we are ready to calculate the rational linking matrix for the graph manifold 
(see Figure 3):
We integrate here over the three dimensional graph manifold 
to obtain a positive definite linking matrix. This manifold has the opposite orientation with respect to the graph manifold 
obtained directly by plumbing of Bh-spheres, which are defined as links of singularities. This construction of the graph manifold 
gives the possibility to represent it also as a link of singularity that guarantees its rational linking matrix to be negative definite (for details see [5] ).
From the tree structure of the graph
, and from the first equation in (13) we immediately obtain, that for 
the nonzero elements are only
for 
If
, we have
Here we use the decomposition (6) of the piece
, and that the integral over trivial Seifert fibration 
is zero. Then according to the two last equations in (13) we obtain the matrix element
(14)
also known as the Chern class of line V-bundle associated with the Seifert fibration of
.
For 
and 
the matrix elements are
(15)
(16)
(17)
Here we have used the decompositions (8) and (9) as well as the notations (4).
4. Conclusions
Comparing the reduced matrix 
(1) with the results (14) and (17) for the rational linking matrix K of the graph manifold 
we observe that decomposing the rational invariants into continued fractions according to (3), (5) and (7), we can create the graph 
(related to diagram
) and obtain the rational linking matrix K of 
just by Gauss-Neumann diagonalization on the Laplacian matrix of
. This is the main result of this article.
We want to conclude with an example of an application of our results for the topological field theory. In [3] [13] we built a set of graph manifolds whose Seifert invariants are constructed on the base of the first 9 prime numbers
. The rational linking matrix of these graph manifolds are positive definite and have diagonal elements (and eigenvalues) simulating the low-energy coupling constants hierarchy of the fundamental interactions of real universe. An example of such matrix is [13]
whose elements are all rational and the diagonal ones are described in (14) and (17) by a sum of three continued fractions. The matrix 
inverse to 
is integer one [6] , the inversion of the rational linking matrices can be done with the help of any program such as MathematicaTM to verify this property, any error on calculation of 
leads to non-integer elements in resulting matrices
. It is also worth mentioning that in the present example the Laplacian block matrix
, corresponding to the matrix
, has 
[13] , while
.
In the 7-dimensional Kaluza-Klein approach to the topological field theory (BF-model), the rational linking matrices of the 3-dimensional graph manifold may be really interpreted as coupling constants matrices [1] . So, the Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices (despite the probably huge rank of the original block matrix
).