A digital linking number for discrete curves

Sébastien Fourey (1) , Rémy Malgouyres (2)

1 Equipe Image - Laboratoire GREYC - UMR6072
2 GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen

A topological invariant, analogous to the linking number as defined in knot theory, is defined for pairs of digital closed paths of Z^3. 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.

Mots clés: Linking number; link; digital homotopy; fundamental group; binary image; topology preservation

Domaines: Topologie géométrique [math.GT]; Mathématique discrète [cs.DM]

Sébastien Fourey, Rémy Malgouyres. A digital linking number for discrete curves. International Journal of Pattern Recognition and Artificial Intelligence, 2001, 15 (7), pp.1053-1074. ⟨10.1142/S0218001401001295⟩. ⟨hal-00338931⟩