In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [19] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [1]. The relation between this new polynomial invariant and the affine index polynomial [14,3] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.
In a recent work of Ayaka Shimizu [5] , she defined an operation named region crossing change on link diagrams, and showed that region crossing change is an unknotting operation for knot diagrams.In this paper, we prove that region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an n-component link diagram if and only if the rank of the associated incidence matrix equals to c − n + 1, here c denotes the size of the graph.
Algebraic homology and cohomology theories for quandles have been studied
extensively in recent years. With a given quandle 2(3)-cocycle one can define a
state-sum invariant for knotted curves(surfaces). In this paper we introduce
another version of quandle (co)homology theory, say positive quandle
(co)homology. Some properties of positive quandle (co)homology groups are given
and some applications of positive quandle cohomology in knot theory are
discussed.Comment: 22 pages, 13 figure
In this paper, from the viewpoint of quandle construction we prove that for any link type L and any diagram of it, there exists another diagram of L such that these two diagrams are Ω 1 -dependent, Ω 2 -dependent and Ω 3 -dependent.
In this paper, we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain how to extend this construction from virtual knots to flat virtual knots.
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