Legendrian non-simple two-bridge knots (Q2322531)

A knot in a 3-dimensional contact manifold is said to be Legendrian if it is everywhere tangent to the contact structure and is said to be transverse if it is everywhere transverse to the contact structure. A knot type is said to be Legendrian simple if any two Legendrian realizations of it with equal Thurston-Bennequin and rotation number are Legendrian isotopic and is said to be transverse simple if any two transverse realizations of it with equal self-linking number are transversely isotopic.\par The two main theorems of this paper provide sufficient conditions for a two-bridge knot type to fail to be either Legendrian or transverse simple. The conditions are expressed in terms of the continued fraction expansion of the rational number that classifies the knot type. Additionally the author proves that certain other two-bridge knots that are not covered by these theorems are not simple. This paper generalizes results of \textit{P. Ozsváth} and \textit{A. I. Stipsicz} [J. Inst. Math. Jussieu 9, No. 3, 601--632 (2010; Zbl 1204.57011)].