On Generalization of Homotopy of Words and Its Applications

TL;DR

This paper generalizes the concept of homotopy in the topology of words and phrases, providing geometric interpretations and extending homotopy invariants to a broader class of combinatorial objects.

Contribution

It introduces a generalized homotopy notion for words and phrases, along with geometric meanings and extended invariants within this framework.

Findings

Generalized homotopy of words and phrases is established.

Geometric interpretations of the generalized homotopy are provided.

Extended homotopy invariants for nanowords are developed.

Abstract

V. Turaev introduced the theory of topology of words and phrases in 2005. This is a combinatorialy extension of the theory of virtual knots and links. In this paper we generalize the notion of homotopy of words and phrases and we give geometric meanings of the generalized homotopy of words. Moreover using the generalized homotopy theory of words and phrases, we extend some homotopy invariants of nanophrases to $S$-homotopy invariant of nanowords with some homotopy data $S$.