A topological invariant, analogous to the linking number as defined in knot theory, is defined for pairs of digital closed paths of ℤ3. This kind of invariant is very useful for proofs which involve homotopy classes of digital paths. Indeed, it can be used, for example, in order to state the connection between the tunnels in an object and the ones in its complement. Even if its definition is not as immediate as in the continuous case it has the good property that it is immediately computable from the coordinates of the voxels of the paths with no need of a regular projection. The aim of this paper is to state and prove that the linking number has the same property as its continuous analogue: it is invariant under any homotopic deformation of one of the two paths in the complement of the other.

As its analogue in the continuous framework, the digital fundamental group represents major information on the topology of discrete objects. However, the fundamental group is abstract information and cannot directly be encoded in a computer by only using its definition. A classical mathematical way to encode a discrete group is to find a presentation of this group. In this paper, we construct a presentation for the fundamental group of an arbitrary graph, and a finite presentation (hence encodable in the memory of a computer) of any subset of ℤ3. This presentation can be computed by an efficient algorithm.

In Ref. 6, two similar characterizations of discrete surfaces of ℤ3 are proposed which are called strong 18-surfaces and strong 26-surfaces. The proposed characterizations consist in some natural global properties of surfaces. In this paper, we first give local necessary conditions for an object to be a strong 26-surface. An object satisfying these local properties is called a near strong 26-surface. Then we construct continuous analogs for near strong 26-surfaces and, using the continuous Jordan Theorem, we prove that the necessary local conditions previously introduced in fact give a complete local characterization of strong 26-surfaces: the class of near strong 26-surfaces coincides with the class of strong 26-surfaces.