The components of the singular locus of a component of a Springer fiber over x² = 0

For $x\in End(K^n)$ satisfying $x^2 = 0$ let $F_x$ be the variety of full flags stable under the action of $x$ (Springer fiber over $x$). The full classification of the components of $F_x$ according to their smoothness was provided in a paper of Fresse-Melnikov in terms of both Young tableaux and link patterns. Moreover in a paper of Fresse the purely combinatorial algorithm to compute the singular locus of a singular components of $F_x$ is provided. However this algorithm involves the computation of the graph of the component, and the complexity of computations grows very quickly, so that in practice it is impossible to use it. In this paper, we construct another algorithm, derived from the algorithm of Fresse, providing all the components of the singular locus of a singular component of $F_x$ in terms of link patterns constructed straightforwardly from its link pattern.