Braided open book decompositions in \(S^{3}\). (English) Zbl 1539.57003
Open books are an important topic in low dimensional topology and play an important role in the study of geometry and topology of \(3\)-manifolds. In the paper under review, the author studies four different ways in which an open book decomposition of the 3-sphere can be defined to be braided. These include generalised exchangeability defined by H. R. Morton and M. Rampichini [Ser. Knots Everything 24, 335–346 (2000; Zbl 0998.57014)] and mutual braiding defined by L. Rudolph [Contemp. Math. 78, 657–673 (1988; Zbl 0669.57004)], which were shown to be equivalent by M. Rampichini [J. Knot Theory Ramifications 10, No. 5, 739–761 (2001; Zbl 1002.57016)], as well as P-fiberedness and a property related to simple branched covers of \(S^{3}\) inspired by work of J. M. Montesinos-Amilibia and H. R. Morton [Proc. Lond. Math. Soc. (3) 62, No. 1, 167–201 (1991; Zbl 0673.57010)]. The author proves that these four notions of a braided open book are actually all equivalent to each other, and shows that all open books in the 3-sphere whose bindings have braid index of at most 3 can be braided in this sense. Also, the author relates his findings to a conjecture on real algebraic links by R. Benedetti and M. Shiota [Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 1, No. 3, 585–609 (1998; Zbl 0910.57010)] and to a stronger version of Harer’s conjecture due to Montesinos and Morton [loc. cit.].
Reviewer: Kun Du (Lanzhou)
MSC:
| 57K10 | Knot theory |
| 57K35 | Other geometric structures on 3-manifolds |
| 14P25 | Topology of real algebraic varieties |
| 32S55 | Milnor fibration; relations with knot theory |
Keywords:
open book decomposition; fibered link; braided surface; P-fibered braid; exchangeable braid; real algebraic link; Hopf plumbingReferences:
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