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Heegaard splittings of the Brieskorn homology spheres that are equivalent after one stabilization. (English) Zbl 1026.57002

In [Geom. Dedicata 76, 229-251 (1999; Zbl 0932.57020)], H. J. Song, Y. H. Im and K. H. Ko introduced within crystallization theory [see, for example, M. Ferri, C. Gagliari and L. Grasselli, Aequationes Math. 31, 121-141 (1986; Zbl 0623.57012)] linear-cut-and-glue (LCG) moves, which are graph-theoretical versions of geometrical \(T\)-transformations on Heegaard diagrams [see J. Singer, Trans. Am. Math. Soc. 35, 88-111 (1933; Zbl 0006.18501)].
In the present paper, a construction by M. Ferri [Proc. Am. Math. Soc. 73, 271-276 (1979; Zbl 0397.57001)] is used to produce, for every \(q \geq 7\), a 3-manifold \(M_q\) which is a 2-fold branched covering of a knot \(K_q\), not equivalent to the torus knot \(T(3,q)\). Then, LCG moves are applied to transform into each other the (genus two) crystallizations corresponding to 2-fold coverings of \(K_q\) and \(T(3,q)\), respectively; this proves \(M_q\) to be homeomorphic to the Brieskorn homology sphere \(\Sigma(2,3,q),\) where \((2,3,q)\) are relatively prime (see [J. Milnor, Knots, Groups, 3-Manif.; pap. dedic. mem. R. H. Fox, 175-225 (1975; Zbl 0305.57003)]).
Note that the previous result extends to \(q \geq 7\) the analogue one, for \(q=7\), contained in [Song, Im and Ko (loc. cit.)] and arising from a famous counterexample of J. S. Birman, F. Gonzalez-Acuna and J. M. Montesinos [Mich. Math. J. 23, 97-103 (1976; Zbl 0321.57004)].

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
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