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http://dx.doi.org/10.4134/JKMS.2010.47.3.585

COMPLEXITY, HEEGAARD DIAGRAMS AND GENERALIZED DUNWOODY MANIFOLDS  

Cattabriga, Alessia (DEPARTMENT OF MATHEMATICS UNIVERSITY OF BOLOGNA)
Mulazzani, Michele (DEPARTMENT OF MATHEMATICS UNIVERSITY OF BOLOGNA)
Vesnin, Andrei (SOBOLEV INSTITUTE OF MATHEMATICS)
Publication Information
Journal of the Korean Mathematical Society / v.47, no.3, 2010 , pp. 585-598 More about this Journal
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.
Keywords
complexity of 3-manifolds; Heegaard diagrams; Dunwoody manifolds; cyclic branched coverings;
Citations & Related Records

Times Cited By Web Of Science : 3  (Related Records In Web of Science)
Times Cited By SCOPUS : 5
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1 M. J. Dunwoody, Cyclic presentations and 3-manifolds, Groups-Korea ’94 (Pusan), 47-55, de Gruyter, Berlin, 1995.
2 L. Grasselli and M. Mulazzani, Genus one 1-bridge knots and Dunwoody manifolds, Forum Math. 13 (2001), no. 3, 379-397.   DOI   ScienceOn
3 L. Grasselli and M. Mulazzani, Seifert manifolds and (1, 1)-knots, Sibirsk. Mat. Zh. 50 (2009), no. 1, 28-39
4 L. Grasselli and M. Mulazzani, Seifert manifolds and (1, 1)-knots, Siberian Math. J. 50 (2009), no. 1, 22-31.   DOI
5 P. Heegaard, Sur l’ “Analysis situs”, Bull. Soc. Math. France 44 (1916), 161-242.
6 M. Mulazzani, All Lins-Mandel spaces are branched cyclic coverings of $S^{3}$, J. Knot Theory Ramifications 5 (1996), no. 2, 239-263.   DOI   ScienceOn
7 M. Mulazzani, A “universal” class of 4-coloured graphs, Rev. Mat. Univ. Complut. Madrid 9 (1996), no. 1, 165-195.
8 C. Petronio and A. Vesnin, Two-sided bounds for the complexity of cyclic branched coverings of two-bridge links, preprint, arXiv:math.GT/0612830v2.
9 L. Neuwirth, An algorithm for the construction of 3-manifolds from 2-complexes, Proc. Cambridge Philos. Soc. 64 (1968), 603-613.   DOI
10 P. Orlik, Seifert Manifolds, Lecture Notes in Mathematics, Vol. 291. Springer-Verlag, Berlin-New York, 1972.
11 A. Cattabriga and M. Mulazzani, Representations of (1, 1)-knots, Fund. Math. 188 (2005), 45-57.   DOI
12 R. C. Randell, The homology of generalized Brieskorn manifolds, Topology 14 (1975), no. 4, 347-355.   DOI   ScienceOn
13 H. Aydin, I. Gultekin, and M. Mulazzani, Torus knots and Dunwoody manifolds, Sibirsk. Mat. Zh. 45 (2004), no. 1, 3-10
14 H. Aydin, I. Gultekin, and M. Mulazzani, Torus knots and Dunwoody manifolds, Siberian Math. J. 45 (2004), no. 1, 1-6.
15 M. Barnabei and L. B. Montefusco Circulant recursive matrices, Algebraic combinatorics and computer science, 111-127, Springer Italia, Milan, 2001.
16 M. R. Casali, Estimating Matveev’s complexity via crystallization theory, Discrete Math. 307 (2007), no. 6, 704-714.   DOI   ScienceOn
17 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.   DOI
18 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.   DOI
19 A. Cavicchioli, On some properties of the groups G(n, l), Ann. Mat. Pura Appl. (4) 151 (1988), 303-316.   DOI
20 A. Kawauchi, A Survey of Knot Theory, Birkhauser, Basel, 1996.
21 S. Matveev, Complexity theory of three-dimensional manifolds, Acta Appl. Math. 19 (1990), no. 2, 101-130.
22 S. Matveev, Algorithmic Topology and Classification of 3-manifolds, Algorithms and Computation in Mathematics, 9. Springer-Verlag, Berlin, 2003.
23 S. Matveev, Recognition and tabulation of 3-manifolds, Dokl. Math. 71 (2005), 20-22.
24 J. Mayberry and K. Murasugi, Torsion-groups of abelian coverings of links, Trans. Amer. Math. Soc. 271 (1982), no. 1, 143-173.   DOI
25 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.
26 J. Minkus, The branched cyclic coverings of 2 bridge knots and links, Mem. Amer. Math. Soc. 35 (1982), no. 255, 1-68.
27 S. Matveev, Tabulations of 3-manifolds up to complexity 12, available from www.topology.kb.csu.ru/recognizer.
28 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.