Geodesic
An Invariant of Links in the Thickened Torus
Matematičeskie zametki, Tome 90 (2011) no. 2, pp. 242-253

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An invariant of links with two and more components in the thickened torus is constructed; the invariant depends on several variables. The construction uses Kauffman's formal theory, which is based on Dehn's representation of knot groups. This invariant is a natural generalization of a polynomial $z$ constructed by Zenkina and Manturov.
Keywords: link in the thickened torus, virtual knot, invariant of links, link diagram, regular link diagram, shadow of a link diagram, Kaufmann's formal theory, Dehn's presentation. 
@article{MZM_2011_90_2_a6,
     author = {M. V. Zenkina},
     title = {An {Invariant} of {Links} in the {Thickened} {Torus}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {242--253},
     publisher = {mathdoc},
     volume = {90},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_2_a6/}
}
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M. V. Zenkina. An Invariant of Links in the Thickened Torus. Matematičeskie zametki, Tome 90 (2011) no. 2, pp. 242-253. http://geodesic.mathdoc.fr/item/MZM_2011_90_2_a6/