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

TLDR: In this paper, the linking matrix of 3D graph manifolds is derived from the Laplacian block matrices by means of Gauss partial diagonalization, a procedure described explicitly by W. Neumann.

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.