Narrow Results

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.

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.

We prove that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams. In turn, the invariant cycles are determined by quasi-indices on the crossings of the diagrams. The found connection between the parities and the (quasi)-indices allows one to define a new series of parities on virtual knots.