Johnson, Jesse; Thompson, Abigail On tunnel number one knots that are not \((1, n)\). (English) Zbl 1218.57008 J. Knot Theory Ramifications 20, No. 4, 609-615 (2011). Summary: We show that the bridge number of a tunnel number \(t\) knot in \(S^3\) with respect to an unknotted genus \(t\) surface is bounded below by a function of the distance of the Heegaard splitting induced by the \(t\) tunnels. It follows that for any natural number \(n\), there is a tunnel number one knot in \(S^3\) that is not \((1,n)\). Cited in 10 Documents MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) Keywords:tunnel number one; bridge number; curve complex × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] DOI: 10.1016/S0040-9383(00)00033-1 · Zbl 0985.57014 · doi:10.1016/S0040-9383(00)00033-1 [2] Kobayashi T., J. Reine Angew. Math. 592 pp 63– [3] DOI: 10.1007/s002220050343 · Zbl 0941.32012 · doi:10.1007/s002220050343 [4] DOI: 10.4310/CAG.1997.v5.n3.a1 · Zbl 0890.57025 · doi:10.4310/CAG.1997.v5.n3.a1 [5] DOI: 10.1017/S0305004100074028 · Zbl 0866.57004 · doi:10.1017/S0305004100074028 [6] DOI: 10.1016/0040-9383(96)00010-9 · Zbl 0867.57009 · doi:10.1016/0040-9383(96)00010-9 [7] DOI: 10.1016/S0166-8641(98)00063-7 · Zbl 0935.57022 · doi:10.1016/S0166-8641(98)00063-7 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.