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.

REFERENCES

1.   Drobotukhina Yu.V. An analogue of the Jones polynomial for links in $RP^3$ and a generalization of the Kauffman-Murasugi theorem. Leningrad Math. J., 1991, vol. 2, no. 3, pp. 613-630.

2.   Miyazaki Katura. Conjugation and prime decomposition of knots in closed, oriented $3$-manifolds. Trans. Amer. Math. Soc., 1989, vol. 313, no. 2, pp. 785-804. doi: 10.1090/S0002-9947-1989-0997679-2.

3.   Gabrovsek Bostjan. Tabulation of prime knots in lens spaes. Mediterr. J. Math., 2017, vol. 14, no. 2, Art. 88,  24 p. doi: 10.1007/s00009-016-0814-5.

4.   Costantino F.   Colored Jones invariants of links in $\#_kS^2\times S^1$ and the volume conjecture.  J. Lond. Math. Soc. (2),  2007, vol. 76, no. 2, pp. 1-15. doi: 10.1112/jlms/jdm029.

5.   Kuperberg G.  What is a virtual link?  Algebr. Geom. Topol., 2003, vol. 3,  pp. 587-591.

6.   Matveev  S.V.  Decomposition of homologically trivial knots in F?IF?I”. 2010,  Dokl. Math., vol. 82, no. 1, pp. 511-513. doi: 10.1134/S1064562410040034.

7.   Turaev V.  Cobordism of knots on surfaces.  J. Topol., 2008, vol. 1, no. 2, pp. 285-305. doi: 10.1112/jtopol/jtn002.

8.   Adams Colin,  Fleming Thomas,  Levin Michael,  Turner Ari M. Crossing number of alternating knots in $S \times I$. Pacific J. Math., 2002, vol. 203,  no. 1, pp.1-22.

9.   Kauffman L.H. State models and the Jones polynomial. Topology,  1987,  vol. 26, no. 3,  pp. 395-407. doi: 10.1016/0040-9383(87)90009-7.

10.   Dye H. A., Kauffman, Louis H.  Minimal surface representations of virtual knots and links.   Algebr. Geom. Topol., 2005,  vol. 5. pp. 509-535. doi: 10.2140/agt.2005.5.509.

11.   Akimova A.A.,  Matveev S.V.  Classification of genus 1 virtual knots having at most five classical crossings.  J. Knot Theory Ramifications,  2014, vol. 23, no. 6, 1450031, 19 p. doi: 10.1142/S021821651450031X.

12.   Akimova A.A., Matveev S.V.  J. Math. Sci., 2014, vol. 202, no. 1, pp. 1-12. doi: 10.1007/s10958-014-2029-2.

13.   Akimova A.A. Classification of knots in the thickened torus with minimal diagrams which are not in a circule and have five crossings.  Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2013, vol. 5, no. 1, pp. 8-11 (in Russian).

14.   Akimova A.A. Classification of knots in a thickened torus with minimal octahedron diagrams which are not  contained in an annulus.  Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 2015, vol. 7, no. 1, pp. 5-10 (in Russian).

15.   Matveev S.V.  Prime decompositions of knots in $T\times I$ .  Topol.  Appl.,  2011,  vol. 159, no. 7,  pp. 1820-1824.  doi: 10.1016/j.topol.2011.04.022.

16.   Korablev Ph., Matveev S.V.  Reductions of knots in thickened surfaces and virtual knots.  Dokl. Math.,  2011,  vol. 83, no. 2,  pp. 262-264. doi: 10.1134/S1064562411020414.

17.   SnapPy 2.5.4 documentation [e-recource]. Available at: http://www.math.uic.edu/t3m/SnapPy.

18.   Burton B. The pachner graph and the simplification of 3-sphere triangulations. Proc. of the Twenty-seventh annual symposium on computational geometry (SCG'11), N. Y., ACM,  2011,  pp. 153 -162. doi: 10.1145/1998196.1998220.

19.   Atlas of 3-Manifolds [site],  Laboratory of Quantum Topology of Chelyabinsk State University. Available at: http://www.matlas.math.csu.ru.