Geodesic
Infinite series of Kishino type knots
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 11 (2014), pp. 975-980.

Voir la notice de l'article provenant de la source Math-Net.Ru

We construct an infinite series of nontrivial virtual knots $\mathcal{K}_n$, $n \geqslant 2$. Each knot in this series is a connected sum of trivial virtual knots. We prove that for each $n$ the genus of $\mathcal{K}_n$ is equal to $n$. As a consequence, two knots $\mathcal{K}_i$ and $\mathcal{K}_j$ are non-equivalent iff $i\neq j$.
Keywords: Kishino knot, knot in thickened surface, virtual knot, genus of the knot. 
@article{SEMR_2014_11_a46,
     author = {Ph. G. Korablev},
     title = {Infinite series of {Kishino} type knots},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {975--980},
     publisher = {mathdoc},
     volume = {11},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2014_11_a46/}
}
TY  - JOUR
AU  - Ph. G. Korablev
TI  - Infinite series of Kishino type knots
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2014
SP  - 975
EP  - 980
VL  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2014_11_a46/
LA  - ru
ID  - SEMR_2014_11_a46
ER  - 
%0 Journal Article
%A Ph. G. Korablev
%T Infinite series of Kishino type knots
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2014
%P 975-980
%V 11
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2014_11_a46/
%G ru
%F SEMR_2014_11_a46
Ph. G. Korablev. Infinite series of Kishino type knots. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 11 (2014), pp. 975-980. http://geodesic.mathdoc.fr/item/SEMR_2014_11_a46/

[1] L. H. Kauffman, “Virtual Knot Theory”, European Journal of Combinatorics, 20:7 (1999), 663–690 | DOI | MR | Zbl

[2] J. S. Carter, S. Kamada, M. Saito, “Stable equivalences of knots on surfaces and virtual knot cobordisms”, Journal of Knot Theory and Its Ramifications, 11:3 (2002), 311–322 | DOI | MR | Zbl

[3] T. Kishino, S. Satoh, “A note on classical knot polynomials”, Journal of Knot Theory and its Ramifications, 13:7 (2004), 845–856 | DOI | MR | Zbl

[4] G. Kuperberg, What is a Virtual Link?, Algebraic and Geometric Topology, 3:20 (2003), 587–591 | DOI | MR | Zbl

[5] H. A. Dye, L. H. Kauffman, “Minimal surface representations of virtual knots and links”, Algebraic Geometric Topology, 5 (2005), 509–535 | DOI | MR | Zbl

[6] A. A. Akimova, S. V. Matveev, “Klassifikatsiya uzlov maloi slozhnosti v utolschennom tore”, Vestnik NGU. Seriya: Matematika, mekhanika, informatika, 12:3 (2012), 10–21 | Zbl

[7] A. Akimova, S. Matveev, “Classification of genus 1 virtual knots having at most five classical crossings”, Journal of Knot Theory and Its Ramifications, 23:6 (2014), 1450031-1–1450031-19 | DOI | MR