Can a hard-sphere fluid feel the topology of a confining pore?

The confinement of simple fluids to narrow pore spaces changes the phase behaviour. A central question is the dependence of thermodynamic properties on the pore shape $K$. The morphometric approach for simple fluids is derived by assuming that the grand potential $\omega(K,\mu,T)$ is an additive functional of $K$. Hadwiger's theorem states that $\omega(K,\mu,T)$ only depends on $K$ as a linear combination of the Minkowski functional, $\Omega=-p(\mu,T) V[K]+\sigma(\mu,T) A[K] + \kappa(\mu,T)C[K]+\overline{\kappa}(\mu,T)X(K)$ where $V$ and $A$ are the volume and interface and $C[K]$ the integrated mean curvature. $X[K]$ is the Euler number that characterises the pore topology. We use density functional theory to demonstrate that this theory is consistent, for the case of triply-periodic network-like pore geometries. For these, the formula $\langle N\rangle(K,\mu,T)=-\partial \Omega/\partial \mu$ can be inverted to give an estimate of $X_f$ deduced from the simulated densities -- the Euler number 'felt' by the fluid. We show that for the Primitive, Gyroid and Diamond minimal surfaces the obtained values are close to $X$. Counter-intuitively, this result suggests that hard sphere fluids can feel topological properties of a confining space, in addition to geometric ones.