Abstract
We prove that an invariant of closed 3-manifolds, called the block number, which is defined via flow-spines, equals the Heegaard genus, except for S 3 and S 2 × S 1. We also show that the underlying 3-manifold is uniquely determined by a neighborhood of the singularity of a flow-spine. This allows us to encode a closed 3-manifold by a sequence of signed labeled symbols. The behavior of the encoding under the connected sum and a criterion for reducibility are studied.
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References
Benedetti R. and Petronio C. (1995). A finite graphic calculus for 3-manifolds. Manuscr. Math. 88(3): 291–310
Benedetti R. and Petronio C. (1997). Branched standard spines of 3-manifolds (Lecture Notes in Math. 1653). Springer, New York
Benedetti R. and Petronio C. (2001). Reidemeister-Turaev torsion of 3-dimensional Euler structures with simple boundary tangency and pseudo-Legendrian knots. Manuscr. Math. 106: 13–74
Casler G. (1965). An imbedding theorem for connected 3-manifolds with boundary. Proc. Amer. Math. Soc. 16: 559–566
Casson A. and Gordon C.McA. (1987). Reducing Heegaard splittings. Top. Appl. 27: 275–283
Endoh M. (2006). On DS-diagrams for 3-manifolds of Heegaard genus 2. Tokyo J. Math. 29: 117–146
Endoh M. and Ishii I. (2005). A new complexity for 3-manifolds. Jpn. J. Math. 31: 131–156
Floyd W. and Oertel U. (1984). Incompressible surfaces via branched surfaces. Topology 23(1): 117–125
Fomenko A. T. and Matveev S. V. (1997). Algorithmic and computer methods for three-manifolds (Math. Appl. 425). Kluwer Academic Publication, Dordrecht
Hempel J. (1976). 3-manifolds (Ann. of Math. Stud. 86). Princeton University Press, Princeton
Ikeda H. (1986). DS-diagrams with E-cycle. Kobe J. Math. 3: 103–112
Ishii I. (1986). Flows and spines. Tokyo J. Math. 9: 505–525
Ishii I. (1992). Moves for flow-spines and topological invariants of 3-manifolds. Tokyo J. Math. 15: 297–312
Kauffman L. H. (1999). Virtual knot theory. Eur. J. Combin. 20: 663–691
Koda, Y.: Spines, Heegaard splittings and Reidemeister-Turaev torsion. Tokyo J. Math. (to appear)
Koda, Y.: A Heegaard-type presentation of branched spines and the Reidemeister-Turaev torsion. (submitted)
Matveev S.V. (2003). Algorithmic topology and classification of 3-manifolds (Algorithms Comput. Math. 9). Springer, Berlin
Oertel U. (1984). Incompressible branched surfaces. Invent. Math. 76(3): 385–410
Theis F. (2002). Topological constructions in the o-graph calculus. Math. Nachr. 241: 170–186
Turaev, V.: Topology of words, math.CO/0503683. preprint
Turaev V. and Viro O. (1992). State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31: 865–902
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Koda, Y. Branched spines and Heegaard genus of 3-manifolds. manuscripta math. 123, 285–299 (2007). https://doi.org/10.1007/s00229-007-0097-z
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DOI: https://doi.org/10.1007/s00229-007-0097-z