One trick to cutting cheesecake neatly is keeping the knife (a very sharp knife) in use clean and smooth. To do this, fill a tall container with hot tap water deep enough to cover the blade of your knife. If no container is available, hold the knife under hot tap water. Dip the knife into the hot water, then wipe it on a clean towel before making every cut. The constant dipping and cleaning will prevent chunks of cake from the previous slice from depositing on top of the next slice.