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Studying links via closed braids. III: Classifying links which are closed 3-braids. (English) Zbl 0813.57010

This paper is one of a series in which closed braid representations of links are studied. One goal of the general programme has been to identify alternative moves on braids in place of the Markov moves, to allow a more controlled classification of links up to equivalence in terms of braids up to the prescribed moves [For Part I see Zbl 0724.57001, Part II: Zbl 0722.57001, Part IV: Zbl 0711.57006, Part V: Zbl 0758.57005 and Part VI: Zbl 0739.57002.]
The present paper covers the case of closed 3-braids. The link classification is shown to be given in most cases by braid conjugacy. The exceptions are classified in terms of an extra move on 3-braids. The precursors, for braids with more strings, of moves which can change a braid to a nonconjugate braid, without altering its closure up to equivalence, are discussed.
The classification presented here allows for a systematic comparison with K. Murasugi’s earlier work [On closed 3-braids, Mem. Am. Math. Soc. 151 (1974; Zbl 0327.55001)], and is applied in an analysis of the extent to which knots and links of certain simple classes can admit presentations as closed 3-braids. The techniques used draw on work of D. Bennequin [Astérisque 107-108, 87-161 (1985; Zbl 0573.58022)] and involve detailed inspection of intersections of surfaces naturally related to the closed braid and its axis.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups
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