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

ABSTRACT

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.

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.

Cite this paper

López, F. , Efremov, V. and Magdaleno, A. (2015) Algorithm for Fast Calculation of Hirzebruch-Jung Continued Fraction Expansions to Coding of Graph Manifolds.*Applied Mathematics*, **6**, 1676-1684. doi: 10.4236/am.2015.610149.

López, F. , Efremov, V. and Magdaleno, A. (2015) Algorithm for Fast Calculation of Hirzebruch-Jung Continued Fraction Expansions to Coding of Graph Manifolds.

References

[1] Hirzebruh, F. (1971) Differentiable Manifolds and Quadratic Forms. Marcel Dekker, New York.

[2] Popescu-Pampu, P. (2007) The Geometry of Continued Fractions and the Topology of Surface Singularities. Advanced Studies in Pure Mathematics, 46.

[3] Griguolo, L., Seminara, D., Szabo, R.J. and Tanzini, A. (2007) Black Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory. Nuclear Physics B, 772, 1-24.

http://dx.doi.org/10.1016/j.nuclphysb.2007.02.030

[4] Saveliev, N. (2002) Fukomoto-Furuta Invariants of Plumbed Homology 3-Spheres. Pacific Journal of Mathematics, 205, 465-490.

http://dx.doi.org/10.2140/pjm.2002.205.465

[5] Becerra, F., Efremov, V. and Hernandez, A. (2014) Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions. Applied Mathematics, 5, 1894-1902.

http://dx.doi.org/10.4236/am.2014.513183

[6] 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, 299-344.

http://dx.doi.org/10.1090/S0002-9947-1981-0632532-8

[7] Saveliev, N. (2002) Invariants for Homology 3-Spheres. Springer, Berlin.

http://dx.doi.org/10.1007/978-3-662-04705-7

[8] Wen, X.G. and Zee, A. (1992) Classification of Abelian Quantum Hall States and Matrix Formulation of Topological Fluids. Physical Review B, 46, 2290.

http://dx.doi.org/10.1103/PhysRevB.46.2290

[9] Balachandran, A.P., Chandar, L. and Sathiapalan, B. (1995) Chern-Simons Duality and Quantum Hall Effect. Nuclear Physics B, 443, 465-500.

http://dx.doi.org/10.1016/0550-3213(95)00122-9

[10] Fujita, M., Li, W., Ryu, S. and Takayanagi, T. (2009) Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy. Journal of High Energy Physics, 2009, JHEP06.

http://dx.doi.org/10.1088/1126-6708/2009/06/066

[11] Harvey, J.A., Kutasov, D., Martinec, E.J. and Moore, G. (2001) Localized Tachyons and RG Flows. arXiv: hep-th/0111154v2.

[12] Efremov, V.N., Mitskievich, N.V., Hernandez Magdaleno, A.M. and Serrano Bautista, R. (2005) Topological Gravity on Plumbed V-Cobordism. Classical and Quantum Gravity, 22, 3725.

http://dx.doi.org/10.1088/0264-9381/22/17/022

[13] 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.

http://dx.doi.org/10.1142/S0217751X10048482

[14] 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.

[15] Tegmark, M., Aguirre, A., Rees, J. and Wilczek, F. (2006) Dimensionless Constants, Cosmology and Other Dark Matter. Physical Review D, 73, Article ID: 023505.

http://dx.doi.org/10.1103/PhysRevD.73.023505

[16] Bousso, R. (2007) TASI Lectures on the Cosmological Constant. arXiv: hep-th/0708.4231.

[17] 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

[18] Diamantini, M.C. and Trugenberger, C.A. (2015) Higgsless Superconductivity from Topological Defects in Compact BF Terms. Nuclear Physics B, 891, 401-419.

http://dx.doi.org/10.1016/j.nuclphysb.2014.12.010

[1] Hirzebruh, F. (1971) Differentiable Manifolds and Quadratic Forms. Marcel Dekker, New York.

[2] Popescu-Pampu, P. (2007) The Geometry of Continued Fractions and the Topology of Surface Singularities. Advanced Studies in Pure Mathematics, 46.

[3] Griguolo, L., Seminara, D., Szabo, R.J. and Tanzini, A. (2007) Black Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory. Nuclear Physics B, 772, 1-24.

http://dx.doi.org/10.1016/j.nuclphysb.2007.02.030

[4] Saveliev, N. (2002) Fukomoto-Furuta Invariants of Plumbed Homology 3-Spheres. Pacific Journal of Mathematics, 205, 465-490.

http://dx.doi.org/10.2140/pjm.2002.205.465

[5] Becerra, F., Efremov, V. and Hernandez, A. (2014) Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions. Applied Mathematics, 5, 1894-1902.

http://dx.doi.org/10.4236/am.2014.513183

[6] 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, 299-344.

http://dx.doi.org/10.1090/S0002-9947-1981-0632532-8

[7] Saveliev, N. (2002) Invariants for Homology 3-Spheres. Springer, Berlin.

http://dx.doi.org/10.1007/978-3-662-04705-7

[8] Wen, X.G. and Zee, A. (1992) Classification of Abelian Quantum Hall States and Matrix Formulation of Topological Fluids. Physical Review B, 46, 2290.

http://dx.doi.org/10.1103/PhysRevB.46.2290

[9] Balachandran, A.P., Chandar, L. and Sathiapalan, B. (1995) Chern-Simons Duality and Quantum Hall Effect. Nuclear Physics B, 443, 465-500.

http://dx.doi.org/10.1016/0550-3213(95)00122-9

[10] Fujita, M., Li, W., Ryu, S. and Takayanagi, T. (2009) Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy. Journal of High Energy Physics, 2009, JHEP06.

http://dx.doi.org/10.1088/1126-6708/2009/06/066

[11] Harvey, J.A., Kutasov, D., Martinec, E.J. and Moore, G. (2001) Localized Tachyons and RG Flows. arXiv: hep-th/0111154v2.

[12] Efremov, V.N., Mitskievich, N.V., Hernandez Magdaleno, A.M. and Serrano Bautista, R. (2005) Topological Gravity on Plumbed V-Cobordism. Classical and Quantum Gravity, 22, 3725.

http://dx.doi.org/10.1088/0264-9381/22/17/022

[13] 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.

http://dx.doi.org/10.1142/S0217751X10048482

[14] 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.

[15] Tegmark, M., Aguirre, A., Rees, J. and Wilczek, F. (2006) Dimensionless Constants, Cosmology and Other Dark Matter. Physical Review D, 73, Article ID: 023505.

http://dx.doi.org/10.1103/PhysRevD.73.023505

[16] Bousso, R. (2007) TASI Lectures on the Cosmological Constant. arXiv: hep-th/0708.4231.

[17] 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

[18] Diamantini, M.C. and Trugenberger, C.A. (2015) Higgsless Superconductivity from Topological Defects in Compact BF Terms. Nuclear Physics B, 891, 401-419.

http://dx.doi.org/10.1016/j.nuclphysb.2014.12.010