The higher topological complexity in digital images

Abstract

Y. Rudyak develops the concept of the topological complexity TC(X) defined by M. Farber. We study this notion in digital images by using the fundamental properties of the digital homotopy. These properties can also be useful for the future works in some applications of algebraic topology besides topological robotics. Moreover, we show that the cohomological lower bounds for the digital topological complexity TC(X,κ) do not hold.

Digital homotopic distance between digital functions

“…Higher digital topological complexity is studied by Is and Karaca in [11]. Motivated from the fact that if digital homotopic distance is a generalization of digital TC, it can be predicted that a higher analog of homotopic distance (defined in a similar way as in [16]; see also [3]) can be realized as a generalization of higher digital TC.…”

Certain topological methods for computing digital topological complexity

“…In the digital meaning, we note that the Schwarz genus of a map p is the Schwarz genus of the digital fibrational substitute of p. Moreover, the fact that the Schwarz genus of a digital map is invariant from the chosen fibrational substitute is proved in [Lemma 3.4, [14]].…”

“…Definition 2.3. [14] Let X [0,m] Z be a digital function space of all continuous functions from [0, m] Z to a digitally connected image (X, κ) for any positive integer m. Then topological complexity of digital images…”

Explicit motion planning in digital projective product spaces

“…Karaca and Is [25] define the concept of digital topological complexity in 2018. The notion of the digital higher topological complexity is added to the literature in [21]. As shown in [21] the cohomological lower bound, particularly zero-divisor cuplength property is not valid for the digital topological complexity.…”

“…The notion of the digital higher topological complexity is added to the literature in [21]. As shown in [21] the cohomological lower bound, particularly zero-divisor cuplength property is not valid for the digital topological complexity. The work on digital topology in finite digital image and given counter examples underline the differences between digital topological complexity and Farber's topological complexity [22,23].…”