Reidemeister Moves and Groups

TL;DR

This paper explores free knots, a class of combinatorial objects related to Reidemeister moves, revealing their non-triviality and the existence of invariants that identify minimal representatives within equivalence classes.

Contribution

It introduces the concept of free knots as non-trivial combinatorial objects and discusses invariants that help identify minimal representatives within their equivalence classes.

Findings

Free knots are non-trivial and admit non-trivial cobordism classes.

Existence of invariants that evaluate to the same diagram, akin to free groups.

Potential for minimal representatives within equivalence classes.

Abstract

Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev, who thought all free knots to be trivial). As it turned out, these new objects are highly non-trivial, and even admit non-trivial cobordism classes. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: an element has its minimal representative which "lives inside" any representative equivalent to it.