Free Knos are Not Invertible

The goal of the present paper is to show that free knots and links are in general not invertible: this fact turns out to be surprisingly non-trivial. Free links (also known as homotopy classes of Gauss phrases) were introduced by Turaev [5], and regularly studied by Manturov [3, 4] and Gibson [1]. We first briefly recall the basic definitions from [3]. By a 4-graph we mean a topological space consisting of finitely many components, each of which is either a circle or a finite graph with all vertices having valency four. A 4-graph is framed if for each vertex of it, the four emanating half-edges are split into two sets of edges called (formally) opposite. A unicursal component of a 4-graph is either a free loop component of it or an equivalence class of edges where two edges a,b are called equivalent if there is a sequence of edges a = a0, . . . , an = b and vertices v1, . . . , vn so that ai and ai+1 are opposite at vi+1. As an example of a free graph one may take the graph of a singular link. Analogously to 4-graphs we define long 4-graphs; here we allow two vertices a, b to have valency one (the others having valency four) in such a way that the edges x, y incident to a and b should belong to the same unicursal component in the above sense. One may also think of these two edges of the long 4-graph to be noncompact, i.e., we may think that the vertices of valency one are removed and the ends of the edges are taken to infinity. By a free link we mean an equivalence class of framed 4-valent graphs modulo the following transformations. For each transformation we assume that only one fixed fragment of the graph is being operated on (this fragment is to be depicted) or some corresponding fragments of the chord diagram. The remaining part of the graph or chord diagram are not shown in the picture; the pieces of the chord diagram not containing chords participating in this transformation, are depicted by punctured arcs. The parts of the graph are always shown in a way such that the formal framing (opposite edge relation) in each vertex coincides with the natural opposite edge relation taken from R. The first Reidemeister move is an addition/removal of a loop, see Fig.1 The second Reidemeister move adds/removes a bigon formed by a pair of edges which are adjacent in two edges, see Fig. 2.