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Uniqueness of bridge surfaces for 2-bridge knots. (English) Zbl 1152.57006
Considering a link \(K\) in a closed 3-manifold \(M\) leads to a generalization of bridge presentations for \(K\subset S^3\). Instead of the 2-sphere instrumental in such a presentation one uses a Heegard surface for \(M\) (bridge surface). The question of uniqueness of a Heegard splitting (up to stabilization) then corresponds to the uniqueness of bridge surfaces up to certain trivial changes of the bridge surface and the link. In addition of the stabilization of the Heegard splitting there is a “perturbation move” and a “meridional stabilization” taking account of the link \(K\subset S^3\). The main result is: Every 2-bridge knot in \(S^3\) has a unique bridge sphere, in the sense that any other bridge surface can be obtained by the changes mentioned above and proper isotopy.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
Full Text: DOI arXiv
[1] DOI: 10.1016/S0040-9383(97)00072-4 · Zbl 0926.57016 · doi:10.1016/S0040-9383(97)00072-4
[2] Ozsv, Clay Math. Proc. 5 pp 3– (2006)
[3] DOI: 10.2140/gtm.1999.2.259 · doi:10.2140/gtm.1999.2.259
[4] DOI: 10.1142/S021821650100069X · Zbl 1001.57038 · doi:10.1142/S021821650100069X
[5] DOI: 10.1007/BF01223371 · Zbl 0656.53078 · doi:10.1007/BF01223371
[6] DOI: 10.1007/BF01388971 · Zbl 0602.57002 · doi:10.1007/BF01388971
[7] DOI: 10.1016/0040-9383(68)90027-X · Zbl 0157.54501 · doi:10.1016/0040-9383(68)90027-X
[8] DOI: 10.2140/gt.2000.4.243 · Zbl 0958.57007 · doi:10.2140/gt.2000.4.243
[9] DOI: 10.1016/0166-8641(87)90092-7 · Zbl 0632.57010 · doi:10.1016/0166-8641(87)90092-7
[10] DOI: 10.1007/BF01388469 · Zbl 0538.57004 · doi:10.1007/BF01388469
[11] DOI: 10.1090/S0002-9947-98-01824-8 · Zbl 0892.57009 · doi:10.1090/S0002-9947-98-01824-8
[12] Bonahon, Ann. Sci. ?cole. Norm. Sup. 16 pp 451– (1983)
[13] DOI: 10.1016/0040-9383(95)00055-0 · Zbl 0858.57020 · doi:10.1016/0040-9383(95)00055-0
[14] Otal, Math. Soc. Lecture Note Ser. 95 pp 143– (1985)
[15] Otal, C. R. Acad. Sci. Paris Sr. I Math. 294 pp 553– (1982)
[16] DOI: 10.2140/gt.2001.5.609 · Zbl 1002.57008 · doi:10.2140/gt.2001.5.609
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