Disk-band graphs in the theory of framed tangles

Authors

  • Vladimir M. Nezhinskij St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation; The Herzen State Pedagogical University of Russia, 48, nab. r. Moyki, St. Petersburg, 191186, Russian Federation
  • Maxim V. Petrov The Herzen State Pedagogical University of Russia, 48, nab. r. Moyki, St. Petersburg, 191186, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2024.305

Abstract

A disk-band graph is a smooth compact two-dimensional manifold with boundary, partitioned into handles; the partition contains only index zero and index one handles and imitates the structure of the graph. (Index zero handles are analogues of the vertices of the graph, index one handles are analogues of the graph edges.) A disk-band graph is called spatial if it is a smooth submanifold of three-dimensional Euclidean space. A tangle is usually understood as a smooth compact one-dimensional submanifold of the standard three-dimensional ball that intersects the boundary of the ball orthogonally, only along its boundary, the intersection is contained in the equator. We call a tangle framed if it is equipped with a smooth field of normal straight lines. It is well known that there is a reduction of the problem of isotopic classification of spatial disk-band graphs to the problem of isotopic classification of framed tangles. This work focuses on the application of (abstract) disk-band graphs to study the set of isotopic classes of framed tangles.

Keywords:

disk-band graph, diagram, tangle, transformer, isotopy

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Published

2024-10-15

How to Cite

Nezhinskij, V. M., & Petrov, M. V. (2024). Disk-band graphs in the theory of framed tangles. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 11(3), 489–494. https://doi.org/10.21638/spbu01.2024.305

Issue

Vol. 11 No. 3 (2024)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.