On symplectic capacities and their blind spots

In this paper we settle three basic questions concerning the Gutt-Hutchings capacities. Our primary result settles a version of the recognition question in the negative. We prove that the Gutt-Hutchings capacities together with the volume, do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary. We also establish two independence properties. We prove that, even for star-shaped domains with smooth boundaries, these capacities are independent from the volume. We also prove that the capacities are mutually independent by constructing, for any $j \in \mathbb{N}$, a family of star-shaped domains, with smooth boundary and the same volume, whose capacities are all equal but the $j^{th}$. The constructions underlying these results are not exotic. They are convex and concave toric domains. A key to the progress made here is a significant simplification of the formulae of Gutt and Hutchings for the capacities of such domains which holds under an additional symmetry assumption. This simplification allows us to identify new blind spots of the capacities which are used to construct the desired examples.