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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)