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Đề tài " Cabling and transverse simplicity "

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2, 3)-cable of the (2, 3)-torus knot is not transversely simple and moreover classify the transverse knots in this knot type. This is the first classification of transverse knots in a nontransversely-simple knot type. We also classify Legendrian knots in this knot type and exhibit the first example of a. | Annals of Mathematics Cabling and transverse simplicity By John B. Etnyre and Ko Honda Annals of Mathematics 162 2005 1305-1333 Cabling and transverse simplicity By John B. ETNyRE and Ko Honda Abstract We study Legendrian knots in a cabled knot type. Specifically given a topological knot type K we analyze the Legendrian knots in knot types obtained from K by cabling in terms of Legendrian knots in the knot type K. As a corollary of this analysis we show that the 2 3 -cable of the 2 3 -torus knot is not transversely simple and moreover classify the transverse knots in this knot type. This is the first classification of transverse knots in a non-transversely-simple knot type. We also classify Legendrian knots in this knot type and exhibit the first example of a Legendrian knot that does not destabilize yet its Thurston-Bennequin invariant is not maximal among Legendrian representatives in its knot type. 1. Introduction In this paper we continue the investigation of Legendrian knots in tight contact 3-manifolds using 3-dimensional contact-topological methods. In EH1 the authors introduced a general framework for analyzing Legendrian knots in tight contact 3-manifolds. There we streamlined the proof of the classification of Legendrian unknots originally proved by Eliashberg-Fraser in EF and gave a complete classification of Legendrian torus knots and figure eight knots. In EH2 we gave the first structure theorem for Legendrian knots namely the reduction of the analysis of connected sums of Legendrian knots to that of the prime summands. This yielded a plethora of non-Legendrian-simple knot types. A topological knot type is Legendrian simple if Legendrian knots in this knot type are determined by their Thurston-Bennequin invariant and rotation number. Moreover we exhibited pairs of Legendrian knots in the same topological knot type with the same Thurston-Bennequin and rotation numbers which required arbitrarily many stabilizations before they became Legendrian isotopic