267 research outputs found
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
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
Virtual links are algorithmically recognisable
We prove that there is an algorithm to decide whehter two virtual links are
equivalent or notComment: 4 Page
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
Framed -valent Graph Minor Theory II: Special Minors and New Examples
In the present paper, we proceed the study of framed -graph minor theory
initiated in ``Framed -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
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
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