Destabilizing Heegaard splittings of knot exteriors
It is known that the connected sum of minimal genus Heegaard splittings yields a minimal genus Heegaard splitting. This follows from \textit{W. Haken's} lemma [Stud. in Math. 5 (Studies Modern Topol.), 39--98 (1968; Zbl 0194.24902)]. On the other hand, Gordon asked the following: Can the connected sum of unstabilized Heegaard splittings yield a stabilized Heegaard splitting? (See Problem 3.9 in [\textit{R. Kirby}, Problems in low-dimensional topology, in: Geometric topology. 1993 Georgia international topology conference,1993, Athens, GA, USA. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 2(pt.2), 35--473 (1997; Zbl 0888.57014)].) In this nice paper, the authors give a necessary and sufficient condition for Heegaard splittings of knot exteriors to admit destabilizations. Then they examine the pair of knots \(K_1\) and \(K_2\), which was introduced by \textit{K. Morimoto} [Topology Appl. 64, No.~2, 165--176 (1995; Zbl 0827.57002)], as an example giving degeneration of tunnel number under connected sum. For this pair, the authors show that the Heegaard splitting of the exterior of \(K_1\# K_2\) derived from certain minimal unknotting tunnel systems of \(K_1\) and \(K_2\) is stabilized.
scientific article; zbMATH DE number 5641763
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