[Paper Review] Homotopy invariants of Gauss words

Andrew Gibson|ArXiv.org|Jan 31, 2009

Geometric and Algebraic Topology3 references17 citations

TL;DR

This paper introduces a homotopy invariant $ z $ for Gauss words, proving that not all Gauss words are homotopically equivalent to the empty word—disproving Turaev's conjecture. Using this invariant and the covering-based height invariant, the authors show there are infinitely many homotopy classes of Gauss words, establishing non-triviality of homotopy and distinguishing it from open homotopy.

ABSTRACT

By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, we show that there are an infinite number of equivalence classes of Gauss words under homotopy.

Motivation & Objective

• To disprove Turaev's conjecture that all Gauss words are open homotopically trivial.
• To construct a homotopy invariant $ z $ that detects non-triviality in Gauss words.
• To show that homotopy and open homotopy of Gauss words are distinct relations.
• To demonstrate the existence of infinitely many homotopy classes of Gauss words using the height invariant.
• To interpret the results in terms of virtual knot theory, showing that crossing change and virtual switch do not unknot all virtual knots.

Proposed method

• Define homotopy of Gauss words via combinatorial moves: generalized Reidemeister moves, shift moves, and isomorphisms.
• Introduce the invariant $ z $, derived from Henrich's smoothing invariant, adapted to remain invariant under homotopy.
• Use the covering construction from Turaev to define the height invariant, which is a homotopy invariant.
• Define an open homotopy invariant $ z_o $, analogous to $ z $, to compare open homotopy with homotopy.
• Apply $ z $ to a specific Gauss word $ ABACDCBD $, showing $ z \neq 0 $, thus proving non-triviality.
• Use the height invariant to construct infinite families of mutually non-homotopic Gauss words.

Experimental results

Research questions

• RQ1 Are there Gauss words that are not homotopically equivalent to the empty word?
• RQ2 Is homotopy of Gauss words strictly stronger than open homotopy?
• RQ3 Can the height invariant be used to construct infinitely many homotopy classes of Gauss words?
• RQ4 Does the invariant $ z $ detect non-triviality in Gauss words under homotopy?
• RQ5 What is the relationship between homotopy of Gauss words and virtual knot invariants under crossing change and virtual switches?

Key findings

• The invariant $ z $ is non-trivial for the Gauss word $ ABACDCBD $, with $ z \neq 0 $, disproving Turaev's conjecture.
• The height invariant is a homotopy invariant that allows construction of infinitely many homotopy classes of Gauss words.
• Proposition 5.9 establishes that there are infinitely many homotopy classes of Gauss words.
• The open homotopy invariant $ z_o $ is strictly stronger than $ z $, showing that open homotopy and homotopy are distinct.
• The Gauss word $ ABACDCBD $ has height 0 under homotopy but height 1 under open homotopy, illustrating the distinction.
• Virtual knots cannot be unknotted using only crossing changes and virtual switches, as shown by the non-triviality of homotopy classes.