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On a genus of a closed surface containing a Brunnian link. (English) Zbl 1175.57009
A link in the 3-sphere is called Brunnian if it is nontrivial but any sub-link obtained by removing a single component is trivial. In the paper under review, the author shows that if a Brunnian link of \(n\) components is isotoped so that it is contained in a closed embedded surface in the 3-sphere, then the genus of the surface is greater than \((n+3)/3\). A key lemma for the proof is the existence of a certain essential tangle decomposition for a Brunnian link, which is of interest independently. Using this lemma, the main result is proved via cut-and-paste techniques in a smart way. Also stated is a conjecture that the bound could be sharpened so that the genus of such a surface is greater than just \(n\).
MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
Keywords:
Brunnian link
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References:
[1] H. Brunn, Über Verkettung, Sitzungsberichte der Bayerische Akad. Wiss., MathPhys. Klasse, 22 (1892), 77-99. · JFM 24.0507.01
[2] M. Ozawa, Morse position of knots and closed incompressible surfaces, J. Knot Theory and its Ramifications, 17 (2008), 377-397. · Zbl 1148.57010
[3] M. Scharlemann, Sutured manifolds and generalized Thurston norms, J. Diff. Geometry, 29 (1989), 557-614. · Zbl 0673.57015
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