We present a heart valve flow computation with the Space-Time Slip Interface Topology Change (ST-SI-TC) method and Isogeometric Analysis (IGA). The computation is for a realistic heart valve model with actual contact between the valve leaflets. The ST-SI-TC method integrates the ST-SI and ST-TC methods in the framework of the ST Variational Multiscale (ST-VMS) method. The STVMS method functions as a moving-mesh method, which maintains high-resolution boundary layer representation near the solid surfaces. The ST-TC method was introduced for moving-mesh computation of flow problems with TC, such as contact between the leaflets of a heart valve. It deals with the contact while maintaining highresolution representation near the leaflet surfaces. The ST-SI method was originally introduced to addresses the challenge involved in high-resolution representation of the boundary layers near spinning solid surfaces. The mesh covering a spinning solid surface spins with it, and the SI between that mesh and the rest of the mesh accurately connects the two sides. This maintains the high-resolution representation near solid surfaces. In the context of heart valves, the SI connects the sectors of meshes containing the leaflets, enabling a more effective mesh moving. In that context, the ST-SI-TC method enables high-resolution representation even when the contact is between leaflets that are covered by meshes with SI. It also enables dealing with contact location change or contact and sliding on the SI. With IGA, in addition to having a more accurate representation of the surfaces and increased accuracy in the flow solution, the element density in the narrow spaces near the contact areas is kept at a reasonable level. Furthermore, because the flow representation in the contact area has a wider support in IGA, the flow computation method becomes more robust. The computation we present for an aortic-valve model shows the effectiveness of the ST-SI-TC-IGA method.