Homotopy Classification of Generalized Phrases in Turaev's Theory of Words

TL;DR

This paper extends Turaev's homotopy classification from words to phrases with up to three letters, introducing a new invariant for nanophrases over any alphabet.

Contribution

It provides a homotopy classification of generalized phrases with up to three letters and introduces a novel invariant for nanophrases applicable to any alphabet.

Findings

Classification of generalized phrases up to three letters

Introduction of a new homotopy invariant for nanophrases

Extension of Turaev's classification to phrases

Abstract

In 2005 V. Turaev introduced the theory of topology of words and phrases. Turaev defined an equivalence relation on generalized words and phrases which is called homotopy. This is suggested by the Reidemeister moves in the knot theory. Then Turaev gave the homotopy classification of generalized words with less than or equal to five letters. In this paper we give the classification of generalized phrases up to homotopy with less than or equal to three letters. To do this we construct a new homotopy invariant for nanophrases over any $\alpha$.