Yes, in a diamond, each carbon atom is indeed covalently bonded to four other carbon atoms. This forms a three-dimensional network structure, often referred to as a crystal lattice. Each carbon atom in the diamond is surrounded by a tetrahedral arrangement of four other carbon atoms, with strong covalent bonds formed between them.
When a diamond is cut, the carbon atoms at the cut surface will still maintain their covalent bonds to four neighboring carbon atoms. However, the bonds on the cut surface will be interrupted, resulting in exposed carbon atoms with dangling bonds. These exposed carbon atoms are considered to be "unsaturated" because they are not bonded to the full four atoms like the interior carbon atoms. As a result, they are more chemically reactive compared to the carbon atoms in the interior of the diamond.
The presence of these unsaturated bonds on the cut surface makes the surface chemically reactive, which is why diamond can be etched or modified through various processes. It is worth noting that the reactivity of the surface atoms can depend on the specific conditions and interactions with other substances.