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.
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.
Cite this paper
López, F. , Efremov, V. and Magdaleno, A. (2014) Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions. Applied Mathematics, 5, 1894-1902. doi: 10.4236/am.2014.513183.
López, F. , Efremov, V. and Magdaleno, A. (2014) Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions. Applied Mathematics, 5, 1894-1902. doi: 10.4236/am.2014.513183.
References
[1] Verlinde, E. (1995) Global Aspects of Electric-Magnetic Duality. Nuclear Physics B, 455, 211-225.
http://dx.doi.org/10.1016/0550-3213(95)00431-Q [2] Efremov, V.N., Mitskievich, N.V., Hernandez Magdaleno, A.M. and Serrano Bautista, R. (2005) Topological Gravity on Plumbed V-Cobordisms. Classical and Quantum Gravity, 22, 3725-3744.
http://dx.doi.org/10.1088/0264-9381/22/17/022 [3] Efremov, V.N., Hernandez Magdaleno, A.M. and Moreno, C. (2010) Topological Origin of the Coupling Constants Hierarchy in Kaluza-Klein Approach. International Journal of Modern Physics A, 25, 2699-2733.
http://dx.doi.org/10.1142/S0217751X10048482 [4] Saveliev, N. (2002) Invariants for Homology3-Spheres. Springer, Berlin, 223.
http://dx.doi.org/10.1007/978-3-662-04705-7 [5] Eisenbud, D. and Neumann, W. (1985) Three-Dimensional Link Theory and Invariants of Plane Curve Singularities. Princeton University Press, Princeton, 172. [6] Saveliev, N. (2002) Fukumoto-Furuta Invariants of Plumbed Homology 3-Spheres. Pacific Journal of Mathematics, 205, 465-490. http://dx.doi.org/10.2140/pjm.2002.205.465 [7] Neumann, W. (1997) Commensurability and Virtual Fibration for Graph Manifolds. Topology, 36, 355-378.
http://dx.doi.org/10.1016/0040-9383(96)00014-6 [8] Hirzebruh, F. (1971) Differentiable Manifolds and Quadratic Forms. Marcel Dekker, New York, 56-58. [9] Neumann, W. (1981) A Calculus for Plumbing Applied to the Topology of Complex Surface Singularities and Degenerating Complex Curves. Transactions of the American Mathematical Society, 268, 299.
http://dx.doi.org/10.1090/S0002-9947-1981-0632532-8 [10] Waldhausen, F. (1967) Eine Klasse Von 3-Dimensionalen Mannigfaltigkeiten. I. Inventiones Mathematicae, 3, 308-333. http://dx.doi.org/10.1007/BF01402956 [11] Popescu-Pampu, P. (2007) The Geometry of Continued Fractions and the Topology of Surface Singularities. In: Brasselet, J.-P. and Suwa, T., Eds., Singularities in Geometry and Topology 2004, Advanced Studies in Pure Mathematics, Vol. 46, 119-195. [12] Beasley, C. and Witten, E. (2005) Non-Abelian Localization for Chern-Simons Theory. Journal of Differential Geometry, 70, 183-323. [13] Efremov, V., Hernandez, A. and Becerra, F. (2014) The Universe as a Set of Topological Fluids with Hierarchy and Fine Tuning of Coupling Constants in Terms of Graph Manifolds. arXiv:1309.0690v2
[1] Verlinde, E. (1995) Global Aspects of Electric-Magnetic Duality. Nuclear Physics B, 455, 211-225.
http://dx.doi.org/10.1016/0550-3213(95)00431-Q [2] Efremov, V.N., Mitskievich, N.V., Hernandez Magdaleno, A.M. and Serrano Bautista, R. (2005) Topological Gravity on Plumbed V-Cobordisms. Classical and Quantum Gravity, 22, 3725-3744.
http://dx.doi.org/10.1088/0264-9381/22/17/022 [3] Efremov, V.N., Hernandez Magdaleno, A.M. and Moreno, C. (2010) Topological Origin of the Coupling Constants Hierarchy in Kaluza-Klein Approach. International Journal of Modern Physics A, 25, 2699-2733.
http://dx.doi.org/10.1142/S0217751X10048482 [4] Saveliev, N. (2002) Invariants for Homology3-Spheres. Springer, Berlin, 223.
http://dx.doi.org/10.1007/978-3-662-04705-7 [5] Eisenbud, D. and Neumann, W. (1985) Three-Dimensional Link Theory and Invariants of Plane Curve Singularities. Princeton University Press, Princeton, 172. [6] Saveliev, N. (2002) Fukumoto-Furuta Invariants of Plumbed Homology 3-Spheres. Pacific Journal of Mathematics, 205, 465-490. http://dx.doi.org/10.2140/pjm.2002.205.465 [7] Neumann, W. (1997) Commensurability and Virtual Fibration for Graph Manifolds. Topology, 36, 355-378.
http://dx.doi.org/10.1016/0040-9383(96)00014-6 [8] Hirzebruh, F. (1971) Differentiable Manifolds and Quadratic Forms. Marcel Dekker, New York, 56-58. [9] Neumann, W. (1981) A Calculus for Plumbing Applied to the Topology of Complex Surface Singularities and Degenerating Complex Curves. Transactions of the American Mathematical Society, 268, 299.
http://dx.doi.org/10.1090/S0002-9947-1981-0632532-8 [10] Waldhausen, F. (1967) Eine Klasse Von 3-Dimensionalen Mannigfaltigkeiten. I. Inventiones Mathematicae, 3, 308-333. http://dx.doi.org/10.1007/BF01402956 [11] Popescu-Pampu, P. (2007) The Geometry of Continued Fractions and the Topology of Surface Singularities. In: Brasselet, J.-P. and Suwa, T., Eds., Singularities in Geometry and Topology 2004, Advanced Studies in Pure Mathematics, Vol. 46, 119-195. [12] Beasley, C. and Witten, E. (2005) Non-Abelian Localization for Chern-Simons Theory. Journal of Differential Geometry, 70, 183-323. [13] Efremov, V., Hernandez, A. and Becerra, F. (2014) The Universe as a Set of Topological Fluids with Hierarchy and Fine Tuning of Coupling Constants in Terms of Graph Manifolds. arXiv:1309.0690v2