A.A. Akimova, S.V. Matveev, V.V. Tarkaev. Classification of links of small complexity in a thickened torus ... P. 18-31

The paper contains the table of links in the thickened torus $T^2\times I$ admitting diagrams with at most four crossings. The links are constructed by a three-step process. First we enumerate all abstract regular graphs of degree 4 with at most four vertices. Then we consider all nonequivalent embeddings of these graphs into $T^2$. After that each vertex of each of the obtained graphs is replaced by a crossing of one of the two possible types, when a segment of the graph lies lower or above another segment. The words "above" and "lower" are understood in the sense of the coordinate of the corresponding point in the interval $I$. As a result, we obtain a family of diagrams of knots and links in $T^2 \times I$. We propose a number of artificial tricks that essentially reduce the enumeration and offer a rigorous proof of the completeness of the table. A generalized version of the Kauffman polynomial is used to prove that all the links are different.

Keywords: link, thickened torus, link table.

The paper was received by the Editorial Office on October 9, 2017.

Alena Andreevna Akimova, Cand. Sci. (Phys.-Math.), South Ural State University, Chelyabinsk,
454080 Russia, e-mail: akimovaaa@susu.ru .

Sergei Vladimirovich Matveev, Dr. Phys.-Math. Sci., Prof., RAS Academician, Krasovskii Institute
of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg,
620990 Russia; Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: matveev@csu.ru .

Vladimir Viktorovich Tarkaev, Cand. Sci. (Phys.-Math.), Chelyabinsk State University, Chelyabinsk,
454001 Russia, e-mail: v.tarkaev@gmail.com.


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