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
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ABSTRACT . 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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    ABSTRACT: Every link diagram can be represented as a signed ribbon graph. However, different link diagrams can be represented by the same ribbon graphs. We determine how checkerboard colourable diagrams of links in real projective space, and virtual link diagrams, that are represented by the same ribbon graphs are related to each other. We also find moves that relate the diagrams of links in real projective space that give rise to (all-A) ribbon graphs with exactly one vertex.
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    ABSTRACT: We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the group of flat virtual braids and the group of flat welded braids, and study some of their properties.
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    ABSTRACT: Y. Miyazawa introduced a two-variable polynomial invariant of virtual knots in 2006 [Magnetic graphs and an invariant for virtual links, J. Knot Theory Ramifications 15 (2006) 1319-1334] and then generalized it to give a multi-variable one via decorated virtual magnetic graph diagrams in 2008. A. Ishii gave a simple state model for the two-variable Miyazawa polynomial by using pole diagrams in 2008 [A multi-variable polynomial invariant for virtual knots and links, J. Knot Theory Ramifications 17 (2008) 1311-1326]. H. A. Dye and L. H. Kauffman constructed an arrow polynomial of a virtual link in 2009 which is equivalent to the multi-variable Miyazawa polynomial [Virtual crossing number and the arrow polynomial, preprint (2008), arXiv: 0810.3858v3,]. We give a bracket model for the multi-variable Miyazawa polynomial via pole diagrams and polar tangles similarly to the Ishii's state model for the two-variable polynomial. By normalizing the bracket polynomial we get the multi-variable Miyazawa polynomial f(K) is an element of Z[A, A(-1), K-1, K-2,...] of a virtual link K. n-similar knots take the same value for any Vassiliev invariant of degree < n. We show that f(K1) = f(K2) mod (A(4) -1)(n) if two virtual links K-1 and K-2 are n-similar. Also we give a necessary condition for a virtual link to be periodic by using n-similarity of virtual tangles and the Miyazawa polynomial.
    Journal of Knot Theory and Its Ramifications 06/2014; 23(07):1460003. DOI:10.1142/S0218216514600037 · 0.44 Impact Factor

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