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Conway products and links with multiple bridge surfaces. (English) Zbl 1158.57011
For a given link in a 3-manifold, it is an interesting problem to compare two bridge surfaces. One can follow the program developed by H. Rubinstein and M. Scharlemann [Topology 35, No. 4, 1005–1026 (1996; Zbl 0858.57020)] to compare distinct Heegaard splittings of a given non-Haken 3-manifold. There the restriction to non-Haken manifolds was introduced to ensure that the relevant Heegaard splittings were strongly irreducible.
In the paper under review, under the analogous condition that the considered bridge surfaces are \(c\)-weakly incompressible, it is shown that, given two different bridge surfaces for a knot, either they can be properly isotoped to intersect in a nonempty collection of curves that are essential (including non-meridional) on both surfaces, or the knot is a Conway product with respect to an incompressible Conway sphere that naturally decomposes both surfaces into bridge surfaces for the respective factor link(s).

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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References:
[1] C. Adams, The knot book. An elementary introduction to the mathematical theory of knots, Amer. Math. Soc., Providence, RI, 2004. · Zbl 1065.57003
[2] D. Bachman and S. Schleimer, Distance and bridge position, Pacific J. Math. 219 (2005), 221–235. · Zbl 1086.57011 · doi:10.2140/pjm.2005.219.221
[3] A. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), 275–283. · Zbl 0632.57010 · doi:10.1016/0166-8641(87)90092-7
[4] J. Cerf, Sur les difféomorphismes de la sphère de dimension trois \((\Gamma_4=0),\) Lecture Notes in Math., 53, Springer-Verlag, Berlin, 1968. · Zbl 0164.24502 · doi:10.1007/BFb0060395
[5] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, 1970 Computational Problems in Abstract Algebra (Conf. Proc., Oxford, 1967), pp. 329–358, Pergamon, Oxford, 1970. · Zbl 0202.54703
[6] C. Gordon and J. Luecke, Knots with unknotting number 1 and essential Conway spheres, Algebr. Geom. Topol. 6 (2006), 2051–2116. · Zbl 1129.57009 · doi:10.2140/agt.2006.6.2051
[7] H. Rubinstein and M. Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds, Topology 35 (1997), 1005–1026. · Zbl 0858.57020 · doi:10.1016/0040-9383(95)00055-0
[8] M. Scharlemann, Proximity in the curve complex: Boundary reduction and bicompressible surfaces, Pacific J. Math. 228 (2006), 325–348. · Zbl 1127.57010 · doi:10.2140/pjm.2006.228.325 · pjm.math.berkeley.edu
[9] M. Scharlemann and A. Thompson, Thin position and Heegaard splittings of the 3-sphere, J. Differential Geom. 39 (1994), 343–357. · Zbl 0820.57005
[10] H. Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954), 245–288. · Zbl 0058.17403 · doi:10.1007/BF01181346 · eudml:169462
[11] J. Schultens, Additivity of bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 135 (2003), 539–544. · Zbl 1054.57011 · doi:10.1017/S0305004103006832
[12] M. Tomova, Multiple bridge surfaces restrict knot distance, Algebr. Geom. Topol. 7 (2007), 957–1006. · Zbl 1142.57005 · doi:10.2140/agt.2007.7.957
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