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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    • "Another way to understand virtual diagrams is to regard them as representatives for oriented Gauss codes [41], [19] [20] (Gauss diagrams). Such codes do not always have planar realizations. "
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    ABSTRACT: This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, studies an extension of the bracket polynomial and the Manturov parity bracket for rotationals. Then the general frameworks for oriented and unoriented quantum invariants are introduced and formulated for rotational virtual links. The paper ends with a section on quantum link invariants in the Hopf algebra framework where one can see the naturality of using regular homotopy combined with virtual crossings (permutation operators), as they occur significantly in the category associated with a Hopf algebra. We show how this approach via categories and quantum algebras illuminates the structure of invariants that we have already described via state summations. In particular, we show that a certain link L has trivial functorial image. This means that this link is not detected by any quantum invariant formulated as outlined in this paper. At this writing, we conjecture that the link L is a non-trivial rotational link. These calculations with the diagrammatic images in quantum algebra show how this category forms a higher level language for understanding rotational virtual knots and links. If the conjecture is true, then there are inherent limitations to studying rotational virtual knots by quantum algebra alone.
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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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    • "An oriented virtual link diagram is a virtual link diagram endowed with an orientation for each component, and two virtual link diagrams are said to be equivalent if they can be related by planar isotopies and a series of generalized Reidemeister moves (r1)–(r3) and (v1)–(v4) depicted in Figure 1. It was proved in [12] that if two classical knot diagrams are equivalent under the generalized Reidemeister moves, then they are equivalent under the classical Reidemeister moves, and consequently this shows that the theory of classical knots embeds into the theory of virtual knots. "
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    ABSTRACT: Given a virtual knot K, we construct a group VG_K called the virtual knot group, and we use the elementary ideals of VG_K to define invariants of K called the virtual Alexander invariants. For instance, associated to the first elementary ideal is a polynomial H_K(s,t,q) in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial G_K(s,t) introduced by Sawollek, Kauffman-Radford, and Silver-Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot K and a representation ρ: VG_K \to GL_n(R), and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial.
    Journal of Knot Theory and Its Ramifications 04/2015; 24(3):62 pp. DOI:10.1142/S0218216515500091 · 0.41 Impact Factor
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