Narrow Results

Free knots are a simplification of virtual knots obtained by forgetting arrow/sign information at classical crossings. First non-trivial examples of free knots were constructed recently by the second named author.

By using parity considerations, we construct invariants of free knots valued in certain groups. These groups have a simple combinatorial description, the first one being the infinite dihedral group.

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 [Topology of words, Proc. Lond. Math. Soc.95(3) (2007) 360–412], who thought all free knots to be trivial). As it turned out, these new objects are highly nontrivial, see [V. O. Manturov, Parity in knot theory, Mat. Sb.201(5) (2010) 65–110], and even admit nontrivial cobordism classes [V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb.203(2) (2012) 45–76]. 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. In this paper, we consider generalizations of free knots by means of (finitely presented) groups. These new objects have lots of nontrivial properties coming from both knot theory and group theory. This connection allows one not only to apply group theory to various problems in knot theory but also to apply Reidemeister moves to the study of (finitely presented) groups. Groups appear naturally in this setting when graphs are embedded in surfaces.

In the present paper, we develop the parity theory invented in [V. O. Manturov, Mathematics201(5) (2010) 693733]; we construct new parities for two-component (virtual and free) links. New parities significantly depend on geometrical properties of diagrams; in particular, they are mutation-sensitive. New parities can be used practically in all problems, where parities were previously applied.

In [1], the authors proved a sliceness criterion for odd free knots: free knots with odd chords. In the present paper, we give a similar criterion for stably odd free knots.

In essence, free knots may be considered as framed 4-graphs. That leads to an important notion of framed 4-graph cobordism and the associated notion of graph genera.

Some additional results on graph and free knot sliceness and cobordism are given.

In [V. O. Manturov, An almost classification of free knots, Dokl. Math.88(2) (2013) 556–558.] the second author constructed an invariant which in some sense generalizes the quantum $\mathrm{\text{sl}}(3)$ link invariant of Kuperberg to the case of free links. In this paper, we generalize this construction to free graph-links. As a result, we obtain an invariant of free graph-links with values in linear combinations of graphs. The main property of this invariant is that under certain conditions on the representative of the free graph-link, we can recover this representative from the value invariant on it. In addition, this invariant allows one to partially classify free graph-links.

A parity is a rule to assign labels to the crossings of knot diagrams in a way compatible with the Reidemeister moves. Parity functors can be viewed as parities which provide to each knot diagram its own coefficient group that contains parities of the crossings. In the paper we describe the universal oriented parity functors for free knots and for knots in a fixed surface.

We prove that the Gaussian parity on free two-dimensional knots is universal.

We consider knot theories possessing parity: each crossing is decreed odd or even according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, then this leads to a possibility of constructing new invariants and proving minimality and non-triviality theorems for knots from these classes, and constructing maps from knots to knots.

Our main example is virtual knot theory and its simplification, free knot theory. By using Gauss diagrams, we show the existence of non-trivial free knots (counterexample to Turaev's conjecture) and construct simple yet deep invariants that rely on parity. Some invariants are valued in graph-like objects and some other are valued in groups.

We discuss applications of parity to virtual knots and ways of extending well-known invariants.

The existence of a non-trivial parity for classical knots remains an open problem.