zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[1] J H Conway, An enumeration of knots and links, and some of their algebraic properties, Pergamon (1970) 329 · Zbl 0202.54703
[2] W B R Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981) 321 · Zbl 0472.57004 · doi:10.2307/1998587
[3] M Scharlemann, Thin position in the theory of classical knots (editors W Menasco, M Thistlethwaite), Elsevier (2005) 429 · Zbl 1097.57013
[4] M Scharlemann, M Tomova, Conway products and links with multiple bridge surfaces, Michigan Math. J. 56 (2008) 113 · Zbl 1158.57011 · doi:10.1307/mmj/1213972401
[5] H Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954) 245 · Zbl 0058.17403 · doi:10.1007/BF01181346 · eudml:169462
[6] J Schultens, Additivity of bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 135 (2003) 539 · Zbl 1054.57011 · doi:10.1017/S0305004103006832
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.