An obstruction to decomposable exact Lagrangian fillings

Abstract: We study some properties of decomposable exact Lagrangian cobordisms between Legendrian links in $\mathbb{R}^3$ with the standard contact structure. In particular, for any decomposable exact Lagrangian filling $L$ of a Legendrian link $K$, we may obtain a normal ruling of $K$ associated with $L$. We prove that the associated normal rulings must have even number of clasps. As a result, we give a particular Legendrian $(4,-(2n+5))$-torus knot, for each $n \geq 0$, which does not have a decomposable exact Lagrangian filling because it has only 1 normal ruling and this normal ruling has odd number of clasps.