On seaweed subalgebras and meander graphs in type D

In 2000, Dergachev and Kirillov introduced subalgebras of “seaweed type” in ${\mathfrak{gl}}_n$ and computed their index using certain graphs, which we call type-A meander graphs. Then the subalgebras of seaweed type, or just “seaweeds”, have been defined by Panyushev (2001) [9] for arbitrary reductive Lie algebras. Recently, a meander graph approach to computing the index in types B and C has been developed by the authors. In this article, we consider the most difficult and interesting case of type $\mathsf{D}$. Some new phenomena occurring here are related to the fact that the Dynkin diagram has a branching node.