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Studying surfaces via closed braids. (English) Zbl 0907.57006

In a series of papers [Pac. J. Math. 154, No. 1, 17-36 (1992; Zbl 0724.57001); Topology Appl. 40, No. 1, 71-82 (1991; Zbl 0722.57001); Pac. J. Math. 161, No. 1, 25-113 (1993; Zbl 0813.57010); Invent. Math. 102, No. 1, 115-139 (1990; Zbl 0711.57006); Trans. Am. Math. Soc. 329, No. 2, 585-606 (1992; Zbl 0758.57005); Pac. J. Math. 156, No. 2, 265-286 (1992; Zbl 0739.57002)] the first author and W. W. Menasco studied closed braid representations of links in \(S^3\) by using “exchange moves” which depend on foliations of spanning surfaces. In the present paper the authors give a solid unified foundation of the techniques used in the above papers by providing a detailed analysis of the (singular) foliation of incompressible surfaces (spanning the link or closed in the link complement). The foliation gives a tiling of the surface and it is shown how the foliations and tilings may be simplified by isotopies of the surfaces. In an example the techniques are applied to show how the foliation of a spanning surface can be used to relate different representatives of a knot. Several interesting open problems are posed.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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