Abstract: A disk-band graph is a smooth compact two-dimensional manifold with a 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 a 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 the 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. © Pleiades Publishing, Ltd. 2024. ISSN 1063-4541, Vestnik St. Petersburg University, Mathematics, 2024, Vol. 57, No. 3, pp. 331–334. Pleiades Publishing, Ltd., 2024. Russian Text The Author(s), 2024, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2024, Vol. 11, No. 3, pp. 489–494.
Original languageEnglish
Pages (from-to)331-334
Number of pages4
JournalVestnik St. Petersburg University: Mathematics
Volume57
Issue number3
DOIs
StatePublished - 1 Sep 2024

    Research areas

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

ID: 126219901