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Link projections and flypes. (English) Zbl 1201.57004

Given a link projection \(\Pi\) in the 2-sphere, the idea of splitting \(\Pi\) into canonical pieces is well-established (see J. H. Conway [Comput. Probl. Abstract Algebra, Proc. Conf. Oxford 1967, 329–358 (1970; Zbl 0202.54703)] and F. Bonahon and L. C. Siebenmann [Math. Ann. 278, 441–479 (1987; Zbl 0629.57007)] for the milestones of the theory); the pieces are known as basic or polyhedral diagrams, and also as rational, algebraic, bretzel, arborescent diagrams, according to different authors.
The present paper takes into account pieces obtained by breaking \(\Pi\) via the so called Haseman circles, i.e. simple closed curves cutting \(\Pi\) transversally in exactly four (simple) points [see M. G. Haseman, On knots, with a census of the amphicheirals with twelve crossings, Trans. Roy. Soc. Edinburgh 52, 235–255 (1918)]. In particular, the main result proves that, for any link projection \(\Pi,\) a suitably defined family of Haseman circles – called minimal Conway family – exists and is unique up to isotopy.
This existence and uniqueness theorem for the canonical decompositions is applied to the classification of Haseman circles and to the localisation of the flypes (for the notion of flype – or distorsion, according to Tait – see [P. G. Tait, Trans. R. Soc. Edinb. 28, 145–190 (1877; JFM 09.0392.09)] and [J. H. Conway, loc. cit.]).

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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