We present a framework for the interactive simulation of surgical cuts such as being practiced in surgical treatment. Unlike most existing methods our framework is based on tetrahedral volume meshes providing more topological flexibility. In order to keep the representation consistent we apply adaptive subdivision schemes dynamically during the simulation. The detection of collisions between the surgical tool and the tissue is accomplished by using an axis aligned bounding box hierarchy, which was adapted for deformable objects. For haptic rendering and feedback, we devised a mechanical scalpel model, which accounts for the most important interaction forces between scalpel and tissue. The relaxation is computed using a localized, semi-implicit ODE solver. The achieved quality and performance of the presented framework is demonstrated using a human soft tissue model.