Let $\Omega$ be a bounded domain of ${\mathbb R}^3$ whose closure $\overline{\Omega}$ is polyhedral, and let ${\mathcal{T}}$ be a triangulation of $\overline{\Omega}$. We devise a fast algorithm for the computation of homological Seifert surfaces of any 1-boundary of ${\mathcal{T}}$, namely, 2-chains of ${\mathcal{T}}$ whose boundary is $\gamma$. Assuming that the boundary of $\Omega$ is sufficiently regular, we provide an explicit formula for a homological Seifert surface of any 1-boundary $\gamma$ of ${\mathcal{T}}$. It is based on the existence of special spanning trees of the complete dual graph and on the computation of certain linking numbers associated with those spanning trees. If the triangulation ${\mathcal{T}}$ is fine, the explicit formula is too expensive to be used directly. To overcome this difficulty, we adopt an easy and very fast elimination procedure, which sometimes fails. In such a case a new unknown can be computed using the explicit formula and the elimination algorithm restarts. The numerical experiments we performed illustrate the efficiency of the resulting algorithm even when the homology of $\Omega$ is not trivial and the triangulation ${\mathcal{T}}$ of $\overline{\Omega}$ consists of millions of tetrahedra.


  1. homological Seifert surfaces
  2. 2-chains with a prescribed boundary
  3. complete dual graph
  4. linking number
  5. spanning trees

MSC codes

  1. 55-04
  2. 68U05
  3. 65D17
  4. 55N99
  5. 52B05

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R. Albanese and G. Rubinacci, Finite element methods for the solution of 3D eddy current problems, Adv. Imaging Electron Phys., 102 (1998), pp. 1--85.
M. Allili and T. Kaczynski, An algorithmic approach to the construction of homomorphisms induced by maps in homology, Trans. Amer. Math. Soc., 352 (2000), pp. 2261--2281, https://doi.org/10.1090/S0002-9947-99-02527-1.
M. Allili and T. Kaczynski, Geometric construction of a coboundary of a cycle, Discrete Comput. Geom., 25 (2001), pp. 125--140, https://doi.org/10.1007/s004540010072.
A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni, and A. Valli, Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3D magnetostatic problems, SIAM J. Numer. Anal., 51 (2013), pp. 2380--2402.
A. Alonso Rodríguez and A. Valli, Eddy Current Approximation of Maxwell Equations, Springer-Verlag Italia, Milan, 2010.
Z. Arai, A rigorous numerical algorithm for computing the linking number of links, Nonlinear Theory Appl., 4 (2013), pp. 104--110.
R. Benedetti, R. Frigerio, and R. Ghiloni, The topology of Helmholtz domains, Expo. Math., 30 (2012), pp. 319--375.
A. Bossavit, Computational Electromagnetism, Academic Press, San Diego, 1998.
M. Brown, Locally flat imbeddings of topological manifolds, Ann. of Math. (2), 75 (1962), pp. 331--341.
J. Cantarella, D. DeTurck, and H. Gluck, Vector calculus and the topology of domains in 3-space, Amer. Math. Monthly, 109 (2002), pp. 409--442.
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, MA, 2009.
T. K. Dey, A. N. Hirani, and B. Krishnamoorthy, Optimal homologous cycles, total unimodularity, and linear programming, SIAM J. Comput., 40 (2011), pp. 1026--1044, https://doi.org/10.1137/100800245.
U. Dierkes, S. Hildebrandt, and F. Sauvigny, Minimal Surfaces, Grundlehren Math. Wiss. 339, Springer, Heidelberg, 2010.
U. Dierkes, S. Hildebrandt, and A. J. Tromba, Global Analysis of Minimal Surfaces, Grundlehren Math. Wiss. 341, Springer, Heidelberg, 2010.
U. Dierkes, S. Hildebrandt, and A. J. Tromba, Regularity of Minimal Surfaces. Grundlehren Math. Wiss. 340, Springer, Heidelberg, 2010.
P. Dłotko and R. Specogna, Physics inspired algorithms for (co)homology computations of three-dimensional combinatorial manifolds with boundary, Comput. Phys. Commun., 184 (2013), pp. 2257--2266, https://doi.org/10.1016/j.cpc.2013.05.006.
J.-G. Dumas, B. D. Saunders, and G. Villard, On efficient sparse integer matrix Smith normal form computations, J. Symbolic Comput., 32 (2001), pp. 71--99, https://doi.org/10.1006/jsco.2001.0451.
N. M. Dunfield and A. N. Hirani, The least spanning area of a knot and the optimal bounding chain problem, in Computational Geometry (SCG'11), ACM, New York, 2011, pp. 135--144, https://doi.org/10.1145/1998196.1998218.
P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, Cambridge University Press, New York, 2004.
R. Hiptmair and J. Ostrowski, Generators of $H_1(\Gamma_h,\mathbb Z)$ for triangulated surfaces: Construction and classification, SIAM J. Comput., 31 (2002), pp. 1405--1423, https://doi.org/10.1137/S0097539701386526.
C. S. Iliopoulos, Worst-case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix, SIAM J. Comput., 18 (1989), pp. 658--669.
T. Kaczynski, Recursive coboundary formula for cycles in acyclic chain complexes, Topol. Methods Nonlinear Anal., 18 (2001), pp. 351--371.
T. Kaczyński, M. Mrozek, and M. Ślusarek, Homology computation by reduction of chain complexes, Comput. Math. Appl., 35 (1998), pp. 59--70, https://doi.org/10.1016/S0898-1221(97)00289-7.
M. Mazur and J. Szybowski, Algebraic construction of a coboundary of a given cycle, Opuscula Math., 27 (2007), pp. 291--300.
P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, Oxford, UK, 2003.
J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley, Menlo Park, CA, 1984.
H. R. Parks, Numerical approximation of parametric oriented area-minimizing hypersurfaces, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 499--511, https://doi.org/10.1137/0913027.
H. R. Parks and J. T. Pitts, Computing least area hypersurfaces spanning arbitrary boundaries, SIAM J. Sci. Comput., 18 (1997), pp. 886--917, https://doi.org/10.1137/S1064827594278903.
D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, 1976.
H. Seifert and W. Threlfall, A Textbook of Topology, Academic Press, New York, 1980.
J. M. Sullivan, A Crystalline Approximation Theorem for Hypersurfaces, Ph.D. thesis, Department of Mathematics, Princeton University, Princeton, NJ, 1990.
J. P. Webb and B. Forghani, A single scalar potential method for 3D magnetostatics using edge elements, IEEE Trans. Magn., 25 (1989), pp. 4126--4128.

Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 1159 - 1187
ISSN (online): 1095-7170


Submitted: 15 June 2015
Accepted: 23 January 2017
Published online: 11 May 2017


  1. homological Seifert surfaces
  2. 2-chains with a prescribed boundary
  3. complete dual graph
  4. linking number
  5. spanning trees

MSC codes

  1. 55-04
  2. 68U05
  3. 65D17
  4. 55N99
  5. 52B05



Ana Alonso Rodríguez

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