Free Knots and Groups

We construct a simple invariant of free link valued in a certain group by using parity.Comment: 4 page

Parity and Cobordisms of Free Knots

In the present paper, we construct a simple invariant which provides a sliceness obstruction for {\em free knots}. This obstruction provides a new point of view to the problem of studying cobordisms of curves immersed in 2-surfaces, a problem previously studied by Carter, Turaev, Orr, and others. The obstruction to sliceness is constructed by using the notion of {\em parity} recently introduced by the author into the study of virtual knots and their modifications. This invariant turns out to be an obstruction for cobordisms of higher genera with some additional constraints.Comment: A result on cobordisms of higher genus with some constraints is added; A list of problems is adde

Virtual links are algorithmically recognisable

We prove that there is an algorithm to decide whehter two virtual links are equivalent or notComment: 4 Page

Minimal diagrams of classical knots

We show that if a classical knot diagram satisfies a certain combinatorial condition then it is minimal with respect to the number of classical crossings. This statement is proved by using the Kauffman bracket and the construction of atoms and knots.Comment: References correcte

New Parities and Coverings over Free Knots

In the present paper, we develop the parity theory invented in \cite{ManSb}; 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.Comment: 18 pages, 11 Figure

Reidemeister Moves and Groups

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.Comment: 14 page

Minimal diagrams of classical and virtual links

We prove that a virtual link diagrams satisfying two conditions on the Khovanov homology is minimal, that is, there is no virtual diagram representing the same link with smaller number of crossings. This approach works for both classical and virtual linksComment: 7 Pages, 1 Table, 1 Figur

Free Knots and Parity

We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, 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 simplifaction, {\em free knot theory}. By using Gauss diagrams, we show the existence of non-trivial free knots (counterexample to Turaev's conjecture), and construct simple and deep invariants made out of 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.Comment: 18 pages;12 Figure

Framed 44-valent Graph Minor Theory II: Special Minors and New Examples

In the present paper, we proceed the study of framed $4$-graph minor theory initiated in ``Framed $4$-valent Graph Minor Theory I. Intoduction. Planarity Criterion '' and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in the projective plane.Comment: 12 page

One-Term Parity Bracket For Braids

In previous papers, the author realized the following principle for many knot theories: if a knot diagram is complicated enough then it reproduces itself, i.e., is a subdiagram of any other diagram equivalent to it. This principle is realized by diagram-valued invariants [ ] of knots such that [K]=K. It turns out that in the case of free braids, the same principle can be realized an unexpectedly easy way by a one-term invariant formula.Comment: 13 pages; In version 2, examples concerning Markov theorem are adde