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Bridge number and Conway products. (English) Zbl 1200.57003
By a classical result of H. Schubert, if \(\beta(K)\) is the bridge number of a knot \(K\), then \(\beta - 1\) is additive with respect to the knot sum (or product). M. Scharlemann and M. Tomova [Mich. Math. J. 56, No. 1, 113–144 (2008; Zbl 1158.57011)] constructed a generalized Conway product of two knots such that \(\beta - 1\) is subadditive. This construction is non-unique and connects two knots along two disjoint arcs of either knot in a certain way. In the present work, the above result is complemented by a lower bound on the bridge number of the above product, namely the inequality \(\beta(K) \geq \beta(K_1) - 1\), where \(K\) is a generalized Conway product and \(K_1\) is the so-called distinguished factor. As a prerequisite, for a class of geometrically defined cases, it is proved that a knot and an accompanying Conway sphere can be moved to a standard situation with respect to the canonical height function of the 3-sphere.

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:
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