Computing Hypercrossed Complex Pairings in Digital Images

Abstract

We consider an additive group structure in digital images and introduce the commutator in digital images. Then we calculate the hypercrossed complex pairings which generates a normal subgroup in dimension 2 and in dimension 3 by using 8-adjacencyand 26-adjacency.

Digital topological complexity numbers

“…Rosenfeld [33], Boxer [3,8,9], Karaca [9], and many other researchers had interests in and studies on this topic in the following years. See [27][28][29][30][31] for more studies and details. One of these studies is about homotopy, an important material for algebraic topology, so Kong [26] defined the digital version of the fundamental group of a given digital picture.…”

Compatible adjacency relations for digital products

“…In relation to the study of this topic, several approaches have been used as follows: in case a Cartesian product (or digital product) has a normal adjacency in [11] (or an S-compatible adjacency in [27], or the L S -property [23]) or the L C -property in [21], many works including [6,20,23,26] dealt with digital topological properties of digital products by using a digital fundamental group [4], digital coverings [10,11,25], an automorphism group of a digital covering [18] and a digital k-surface structure [2,3,8,14,31]. Computing Hyper-crossed complex pairings in digital images was studied in [35]. However, these approaches could not be enough to deal with the issue because the other cases remain open.…”