arXiv subjects
Publications and source records attributed to Mengjian Xu.
In this note, we construct a chord index homomorphism from a subgroup of $H_1(Σ, \mathbb{Z})$ to the group of chord indices of a knot $K$ in $Σ\times I$. Some knot invariants derived from this homomorphism are discussed.
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.
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.