Narrow Results

The motivation for this work is to construct a map from classical knots to virtual ones. What we get in the paper is a series of maps from knots in the full torus (thickened torus) to flat-virtual knots. We give definition of flat-virtual knots and presents Alexander-like polynomial and (picture-valued) Kauffman bracket for them.

A parity is a labeling of the crossings of knot diagrams which is compatible with Reidemeister moves. We define the notion of parity for based matrices — algebraic objects are introduced by Turaev in his research on virtual strings. We present the reduced stable parity on based matrix which gives a new example of a parity of virtual knots.