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Holonomic and Legendrian parametrizations of knots. (English) Zbl 1001.57015

Summary: Holonomic parametrizations of knots were introduced in [ibid. 6, No. 1, 115-123 (1997; Zbl 0888.57004)] by V. A. Vassiliev, who proved that every knot type can be given a holonomic parametrization. Our main result is that any two holonomic knots which represent the same knot type are isotopic in the space of holonomic knots. A second result emerges through the techniques used to prove the main result: strong and unexpected connections between the topology of knots and the algebraic solution to the conjugacy problem in the braid groups, via the work of F. A. Garside [Q. J. Math., Oxf. II. Ser. 20, 235-254 (1969; Zbl 0194.03303)]. We also discuss related parametrizations of Legendrian knots, and uncover connections between the concepts of holonomic and Legendrian parametrizations of knots.

MSC:

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

[1] Ad S.I, Matematcheskie Zametki 36 (1) pp 25– (1986)
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