Geodesic
Classification of prime projections of knots in the thickened torus of genus 2 with at most 4 crossings
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 12 (2020) no. 1, pp. 5-13

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

We begin classification of prime knots in the thickened torus of genus 2 having diagrams with at most 4 crossings. To this end, it is enough to construct a table of prime knot projections with at most 4 crossings, and use the table to obtain table of prime diagrams, i. e. table of prime knots. In this paper, we present the result of the first step, i. e. we construct a table of prime projections of knots in the thickened torus of genus 2 having at most 4 crossings. First, we introduce definition of prime projection of a knot in the thickened torus of genus 2. Second, we construct a table of prime projections of knots in the thickened torus of genus 2 having at most 4 crossings. To this end, we enumerate graphs of special type and consider all possible embeddings of the graphs into the torus of genus 2 that lead to prime projections. In order to simplify enumeration of the embeddings, we prove some auxiliary statements. Finally, we prove that all obtained projections are inequivalent. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed table.
Keywords: prime projection, knot, thickened torus of genus 2
Mots-clés : table. 
@article{VYURM_2020_12_1_a0,
     author = {A. A. Akimova},
     title = {Classification of prime projections of knots in the thickened torus of genus 2 with at most 4 crossings},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
     pages = {5--13},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VYURM_2020_12_1_a0/}
}
TY  - JOUR
AU  - A. A. Akimova
TI  - Classification of prime projections of knots in the thickened torus of genus 2 with at most 4 crossings
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
PY  - 2020
SP  - 5
EP  - 13
VL  - 12
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VYURM_2020_12_1_a0/
LA  - en
ID  - VYURM_2020_12_1_a0
ER  - 
%0 Journal Article
%A A. A. Akimova
%T Classification of prime projections of knots in the thickened torus of genus 2 with at most 4 crossings
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
%D 2020
%P 5-13
%V 12
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VYURM_2020_12_1_a0/
%G en
%F VYURM_2020_12_1_a0
A. A. Akimova. Classification of prime projections of knots in the thickened torus of genus 2 with at most 4 crossings. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 12 (2020) no. 1, pp. 5-13. http://geodesic.mathdoc.fr/item/VYURM_2020_12_1_a0/