We say that a cubical 2-knot \(K^{2}\) is an embedding of the 2-sphere in the 2-skeleton of the canonical cubulation of \(\mathbb {R}^4\) ; in particular, \(K^{2}\) is the union of \(m(K^{2})\) unit squares, hence \(m(K^{2})\) is its area. The following natural question arises: Which is the smallest area of a cubical 2-knot to be knotted? In this paper, we prove that if the area of a cubical 2-knot \(K^2\) is smaller than 48, then \(K^2\) is unknotted. To do that, we define a new curvature called positive total curvature, which in this sense, is a generalization of the total curvature for polygonal curves given by Milnor (Ann Math 53(2):248–257, 1950).
