Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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COMPLEXITY, HEEGAARD DIAGRAMS AND GENERALIZED DUNWOODY MANIFOLDS

  • Published : 2010.05.01
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Abstract

We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and Pervova we obtain lower bounds.

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References

  1. H. Aydin, I. Gultekin, and M. Mulazzani, Torus knots and Dunwoody manifolds, Sibirsk. Mat. Zh. 45 (2004), no. 1, 3-10
  2. H. Aydin, I. Gultekin, and M. Mulazzani, Torus knots and Dunwoody manifolds, Siberian Math. J. 45 (2004), no. 1, 1-6.
  3. M. Barnabei and L. B. Montefusco Circulant recursive matrices, Algebraic combinatorics and computer science, 111-127, Springer Italia, Milan, 2001.
  4. M. R. Casali, Estimating Matveev’s complexity via crystallization theory, Discrete Math. 307 (2007), no. 6, 704-714. https://doi.org/10.1016/j.disc.2006.07.021
  5. M. R. Casali and P. Cristofori, Computing Matveev’s complexity via crystallization theory: the orientable case, Acta Appl. Math. 92 (2006), no. 2, 113-123. https://doi.org/10.1007/s10440-006-9065-y
  6. A. Cattabriga and M. Mulazzani, All strongly-cyclic branched coverings of (1, 1)-knots are Dunwoody manifolds, J. London Math. Soc. (2) 70 (2004), no. 2, 512-528. https://doi.org/10.1112/S0024610704005538
  7. A. Cattabriga and M. Mulazzani, Representations of (1, 1)-knots, Fund. Math. 188 (2005), 45-57. https://doi.org/10.4064/fm188-0-3
  8. A. Cavicchioli, On some properties of the groups G(n, l), Ann. Mat. Pura Appl. (4) 151 (1988), 303-316. https://doi.org/10.1007/BF01762801
  9. M. J. Dunwoody, Cyclic presentations and 3-manifolds, Groups-Korea ’94 (Pusan), 47-55, de Gruyter, Berlin, 1995.
  10. L. Grasselli and M. Mulazzani, Genus one 1-bridge knots and Dunwoody manifolds, Forum Math. 13 (2001), no. 3, 379-397. https://doi.org/10.1515/form.2001.013
  11. L. Grasselli and M. Mulazzani, Seifert manifolds and (1, 1)-knots, Sibirsk. Mat. Zh. 50 (2009), no. 1, 28-39
  12. L. Grasselli and M. Mulazzani, Seifert manifolds and (1, 1)-knots, Siberian Math. J. 50 (2009), no. 1, 22-31. https://doi.org/10.1007/s11202-009-0003-x
  13. P. Heegaard, Sur l’ “Analysis situs”, Bull. Soc. Math. France 44 (1916), 161-242.
  14. A. Kawauchi, A Survey of Knot Theory, Birkhauser, Basel, 1996.
  15. S. Matveev, Complexity theory of three-dimensional manifolds, Acta Appl. Math. 19 (1990), no. 2, 101-130.
  16. S. Matveev, Algorithmic Topology and Classification of 3-manifolds, Algorithms and Computation in Mathematics, 9. Springer-Verlag, Berlin, 2003.
  17. S. Matveev, Recognition and tabulation of 3-manifolds, Dokl. Math. 71 (2005), 20-22.
  18. S. Matveev, Tabulations of 3-manifolds up to complexity 12, available from www.topology.kb.csu.ru/recognizer.
  19. S. Matveev, C. Petronio, and A. Vesnin, Two-sided asymptotic bounds for the complexity of some closed hyperbolic three-manifolds, J. Australian Math. Soc., to appear.
  20. J. Mayberry and K. Murasugi, Torsion-groups of abelian coverings of links, Trans. Amer. Math. Soc. 271 (1982), no. 1, 143-173. https://doi.org/10.2307/1998756
  21. J. Milnor, On the 3-dimensional Brieskorn manifolds M(p, q, r), Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 175-225. Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N. J., 1975.
  22. J. Minkus, The branched cyclic coverings of 2 bridge knots and links, Mem. Amer. Math. Soc. 35 (1982), no. 255, 1-68.
  23. M. Mulazzani, All Lins-Mandel spaces are branched cyclic coverings of $S^{3}$, J. Knot Theory Ramifications 5 (1996), no. 2, 239-263. https://doi.org/10.1142/S0218216596000175
  24. M. Mulazzani, A “universal” class of 4-coloured graphs, Rev. Mat. Univ. Complut. Madrid 9 (1996), no. 1, 165-195.
  25. L. Neuwirth, An algorithm for the construction of 3-manifolds from 2-complexes, Proc. Cambridge Philos. Soc. 64 (1968), 603-613. https://doi.org/10.1017/S0305004100043279
  26. P. Orlik, Seifert Manifolds, Lecture Notes in Mathematics, Vol. 291. Springer-Verlag, Berlin-New York, 1972.
  27. C. Petronio and A. Vesnin, Two-sided bounds for the complexity of cyclic branched coverings of two-bridge links, preprint, arXiv:math.GT/0612830v2.
  28. R. C. Randell, The homology of generalized Brieskorn manifolds, Topology 14 (1975), no. 4, 347-355. https://doi.org/10.1016/0040-9383(75)90019-1

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