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BLOBS AND FLIPS ON GEMS by:7 (Source: Crossref)

    In this paper we prove that two n-gems induce the same manifold if and only if they are linked by a finite sequence of gem moves. A gem move is either a blob move, consisting in the creation or cancellation of an n-dipole, or a clean flip, which is a switch of a pair of edges of the same color that thickens an h-dipole, 1 ≤ h ≤ n - 1, or the inverse operation, which slims an h-dipole, 2 ≤ h ≤ n. Moreover we prove that we can reorder the gem moves, so that all the blob creations precede all clean flips which then precede all the blob cancellations. This reordering is of interest because it is an easy matter to decide whether two gems are linked by a finite sequence of clean flips. As a consequence, if a bound for the number of blob creations is established, then there exists a deterministic finite algorithm to decide whether two gems induce the same manifold or not.

    AMSC: Primary 57Q05, Primary 57M15, Secondary 57M12, Secondary 57M25, Secondary 05C10


    • M. R. Casali, Rev. Mat. Univ. Complutense Madr. 10, 129 (1997). Google Scholar
    • C.   Berge , Principles of Combinatorics ( Academic Press , 1971 ) . Google Scholar
    • T. H   Cormen , C. E.   Leiserson and R. L.   Rivest , Introduction to Algorithms ( MIT Press/McGraw-Hill , 1989 ) . Google Scholar
    • M. Ferri and C. Gagliardi, Pacific J. Math. 100, 85 (1982). Crossref, ISIGoogle Scholar
    • M. Ferri, C. Gagliardi and L. Grasselli, Aequationes Math. 31, 121 (1986). CrossrefGoogle Scholar
    • C. Gagliardi, Ann. Univ. Ferrara 33, 51 (1987). CrossrefGoogle Scholar
    • L.   Kauffman and S.   Lins , Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds , Annals of Mathematical Studies   134 ( Princeton University Press , Princeton, NJ , 1994 ) . CrossrefGoogle Scholar
    • W. B. R. Lickorish, Simplicial moves on complexes and manifolds, Geometry and Topology Monographs, Proceedings of the Kirbyfest2 (1999) pp. 299–320. Google Scholar
    • S. Lins and A. Mandel, Discrete Math. 57, 261 (1985). Crossref, ISIGoogle Scholar
    • S.   Lins , Gems, Computers and Attractors for 3-Manifolds , Series on Knots and Everything   5 ( World Scientific , 1995 ) . CrossrefGoogle Scholar
    • S. Lins, Discrete Math. 177, 145 (1997). Crossref, ISIGoogle Scholar
    • S. Lins and J. S. Carter, Adv. Math. 143, 251 (1999). ISIGoogle Scholar
    • A. Mijatovic, Math. Ann. 330, 235 (2004). Crossref, ISIGoogle Scholar
    • A. Mijatovic, Pacific J. Math. 219, 139 (2005). Crossref, ISIGoogle Scholar
    • C.   Rourke and B.   Sanderson , Introduction to Piecewise-Linear Topology ( Springer-Verlag , Berlin, Heildeberg, New York , 1972 ) . CrossrefGoogle Scholar
    • U. Pachner, European. J. Combin. 12, 129 (1991). Crossref, ISIGoogle Scholar
    • A.   Schrijver , Combinatorial Optimization: Polyhedra and Efficiency ( Springer-Verlag , Berlin, Heildeberg, New York , 2003 ) . Google Scholar
    • L. Lins, BLINK: A new tool for manipulating 3-manifolds, PhD Thesis, under preparation (2006) . Google Scholar