$(4,-(2n+5))$-torus knot with only 1 normal ruling

Abstract

The main purpose of this paper is to provide an infinite family of counter examples of the open problem mentioned in [2]. In particular, we present an infinite family of a particular Legendrian $(4,-(2n+5))$-torus knot, for each $n \geq 0$, which has only 1 normal ruling, but do not satisfy the even number of clasps condition of Theorem 3 of [2]. Thus, these normal rulings cannot imply the existence of a decomposable exact Lagrandian filling.