Finite Type Invariants Of Classical And Virtual Knots

Topology (Impact Factor: 0.23). 02/1970; 39(5). DOI: 10.1016/S0040-9383(99)00054-3
Source: arXiv


. We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant and show that the restriction of any such invariant of degree n to classical knots is an invariant of degree n in the classical sense. A universal invariant of degree n is defined via a Gauss diagram formula. This machinery is used to obtain explicit formulas for invariants of low degrees. The same technique is also used to prove that any finite type invariant of classical knots is given by a Gauss diagram formula. We introduce the notion of n-equivalence of Gauss diagrams and announce virtual counter-parts of results concerning classical n-equivalence. 1. Virtualization Recently L. Kauffman introduced a notion of a virtual knot, extending the knot theory in an unexpected direction. We show here that this extension motiva...

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Available from: Michael Polyak, Feb 24, 2014
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    • "Recently some generalizations of classical knots and links were defined and studied: singular links [19] [4], virtual links [10] [17], welded links [7] and fused links [12] [2] [16]. The problem of constructing invariants is also important for all of this knot theories. "
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    ABSTRACT: We construct the complete invariant for fused links. It is proved that the set of equivalence classes of $n$-component fused links is in one-to-one correspondence with the set of elements of the abelization $UVP_n/UVP_n^{\prime}$ up to conjugation by the elements from the symmetric group $S_n<UVB_n$.
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    • "When we allow OC, we obtain the class of welded knotted objects [8] [1], while allowing both forbidden moves yields the notion of fused knotted objects [10]. It is well known that braids and knots embed in their welded counterpart (see [8] and [6], respectively), while this question is still open for (string) links. This is not the case for fused objects, since all fused knots are equivalent to the unknot [9] [16]. "
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    ABSTRACT: We consider the quotient of welded knotted objects under several equivalence relations, generated respectively by self-crossing changes, $\Delta$ moves, self-virtualizations and forbidden moves. We prove that for welded objects up to forbidden moves or classical objects up to $\Delta$ moves, the notions of links and string links coincide, and that they are classified by the (virtual) linking numbers; we also prove that the $\Delta$ move is an unknotting operation for welded (long) knots. For welded knotted objects, we prove that forbidden moves imply the $\Delta$ move, the self-crossing change and the self-virtualization, and that these four local moves yield pairwise different quotients, while they collapse to only two distinct quotients in the classical case.
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    • "3) if two classical braids are equivalent as virtual braids, then they are equivalent as classical braids. The first result [3] and the second result are proved by classical methods (fundamental group). The third one is firstly proved in [2] and in [1] it is proved by parity method . "
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    ABSTRACT: In the present paper, we define an invariant of free links valued in a free product of some copies of $\mathbb{Z}_{2}$. In \cite{Ma2} the second named author constructed a connection between classical braid group and group presentation generated by elements corresponding to horizontal trisecants. This approach does not apply to links nor tangles because it requires that when counting trisecants, we have the same number of points at each level. For general tangles, trisecants passing through one component twice may occur. Free links can be obtained from tangles by attaching two end points of each component. We shall construct an invariant of free links and free tangles valued in groups as follows: we associate elements in the groups with 4-valent vertices of free tangles(or free links). For a free link with enumerated component, we `read' all the intersections when traversing a given component and write them as a group element. The problem of `pure crossings' of a component with itself by using the following statement: {\em if two diagrams with no pure crossings are equivalent then they are equivalent by a sequence of moves where no intermediate diagram has a pure crossing.} This statement is a result of a sort that an equivalence relation within a subset coincides with the equivalence relation induced from a larger set and it is interesting by itself.
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