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

TL;DR

This paper constructs an infinite family of Legendrian torus knots with a unique normal ruling that defies previous conditions, providing counterexamples to a known open problem in Legendrian knot theory.

Contribution

It introduces an infinite family of Legendrian (4,-(2n+5))-torus knots with a single normal ruling that do not meet the even number of clasps condition, challenging existing theorems.

Findings

Existence of Legendrian (4,-(2n+5))-torus knots with only 1 normal ruling

Counterexamples to the even number of clasps condition

Normal rulings do not necessarily imply decomposable exact Lagrangian fillings

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.