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.
A weak chord index Ind ′ is constructed for self crossing points of virtual links. Then a new writhe polynomial W of virtual links is defined by using Ind ′ . W is a generalization of writhe polynomial defined in [6]. Based on W , three invariants of virtual links are constructed. These invariants can be used to detect the non-trivialities of Kishino knot and flat Kishino knot.
We give a brief survey of virtual knot invariants derived from chord parity or chord index. These invariants have grown into an area in its own right due to rapid developing in the last decade. Several similar invariants of flat virtual knots and free knots are also discussed.
In this note, we construct a chord index homomorphism from a subgroup of H 1 (Σ, Z) to the group of chord indices of a knot K in Σ × I. Some knot invariants derived from this homomorphism are discussed.
In this paper, we define transcendental polynomial invariants for two-component virtual string links. One of these invariants is a strictly refinement of the linking polynomial in [M. Xu and H. Gao, Linking polynomials of virtual string links, Sci. China Math. 61(7) (2018) 1287–1302]. It is a homotopy invariant and can distinguish some virtual string links from their mirror images. We also define a transcendental polynomial invariant for two-component flat virtual string links. These invariants can be used to study the periodicity and linking crossing number of virtual string links.
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