Geodesic
Weak parities and functorial maps
Contemporary Mathematics. Fundamental Directions, Topology, Tome 51 (2013), pp. 123-141

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We consider functorial maps and weak parities that are two equivalent descriptions of one object. Functorial maps allow one to transform knots and extend knot invariants with these transformations. We introduce maximal weak parity and calculate it for knots in a given closed oriented surface. The weak parity induce a projection from virtual knots onto classical ones. 
@article{CMFD_2013_51_a7,
     author = {I. M. Nikonov},
     title = {Weak parities and functorial maps},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {123--141},
     publisher = {mathdoc},
     volume = {51},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2013_51_a7/}
}
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I. M. Nikonov. Weak parities and functorial maps. Contemporary Mathematics. Fundamental Directions, Topology, Tome 51 (2013), pp. 123-141. http://geodesic.mathdoc.fr/item/CMFD_2013_51_a7/