Geodesic
Classification of Low Complexity Knots in the Thickened Torus
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 12 (2012) no. 3, pp. 10-21 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We compose the table of knots in the thickened torus $ T\times I$ which have diagrams with $\leq 4$ crossing points. The knots are constructed by a three-step process. First we enumerate regular graphs of degree 4, then for each graph we enumerate all corresponding knot projection, and after that we construct the corresponding minimal diagrams. Several known and new tricks made possible to keep the process within reasonable limits and offer a rigorous theoretical proof of the completeness of the table. For proving that all knots are different we use a generalized version of the Kauffman polynomial.
Keywords: knot, thickened torus, knot table. 
@article{VNGU_2012_12_3_a1,
     author = {A. A. Akimova and S. V. Matveev},
     title = {Classification of {Low} {Complexity} {Knots} in the {Thickened} {Torus}},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {10--21},
     year = {2012},
     volume = {12},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2012_12_3_a1/}
}
TY  - JOUR
AU  - A. A. Akimova
AU  - S. V. Matveev
TI  - Classification of Low Complexity Knots in the Thickened Torus
JO  - Sibirskij žurnal čistoj i prikladnoj matematiki
PY  - 2012
SP  - 10
EP  - 21
VL  - 12
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VNGU_2012_12_3_a1/
LA  - ru
ID  - VNGU_2012_12_3_a1
ER  - 
%0 Journal Article
%A A. A. Akimova
%A S. V. Matveev
%T Classification of Low Complexity Knots in the Thickened Torus
%J Sibirskij žurnal čistoj i prikladnoj matematiki
%D 2012
%P 10-21
%V 12
%N 3
%U http://geodesic.mathdoc.fr/item/VNGU_2012_12_3_a1/
%G ru
%F VNGU_2012_12_3_a1
A. A. Akimova; S. V. Matveev. Classification of Low Complexity Knots in the Thickened Torus. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 12 (2012) no. 3, pp. 10-21. http://geodesic.mathdoc.fr/item/VNGU_2012_12_3_a1/

[1] Tait P. G., “On Knots. I; II; III”, Scientific Papers, v. 1, Cambridge University Press, 1898–1900 ; Trans. Roy. Soc. Edinburgh, 28 (1877), 35–79 | Zbl

[2] Alexander J. W., Briggs G. B., “On Types of Knotted Curves”, Ann. Math., 28 (1927), 562–586 | DOI | MR | Zbl

[3] Rolfsen D., Knots and Links, Publish of Perish, Berkeley, 1976 ; 2003 | MR | Zbl

[4] Hoste J., Thistlethwaite M., Weeks J., “The First 1,701,935 Knots”, Math. Intelligencer, 20:4 (1998), 33–48 | DOI | MR | Zbl

[5] Drobotukhina Yu. V., “Analog polinoma Dzhonsa dlya zatseplenii v $RP^3$ i obobschenie teoremy Kauffmana–Murasugi”, Algebra i analiz, 2:3 (1991), 613–630 | MR | Zbl

[6] Drobotukhina Yu. V., “Classification of Links in $\mathbb{R}P^3$ with at Most Six Crossings”, Advances in Soviet Mathematics, 18:1 (1994), 87–121 | MR | Zbl

[7] Bogdanov A., Meshkov V., Omelchenko A., Petrov M., “Classification of $k$-Tangle Projections Using the Cascade Representation”, J. of Knot Theory and Its Ramifications, 21:7 (2011), 17 pp. | DOI | MR

[8] Grishanov S., Meshkov V., Omelchenko A., “Kauffman-Type Polynomial Invariants for Doubly Periodic Structures”, J. of Knot Theory and Its Ramifications, 16:6 (2007), 779–788 | DOI | MR | Zbl

[9] Kauffman L., “State Models and the Jones Polynomial”, Topology, 26:3 (1987), 395–407 | DOI | MR | Zbl

[10] Prasolov V. V., Sosinskii A. B., Uzly, zatsepleniya, kosy i trekhmernye mnogoobraziya, MTsNMO, 1997