TY - JOUR

T1 - Disk-Band Graphs in the Theory of Framed Tangles

AU - Nezhinskij, V.M.

AU - Petrov, M.V.

N1 - Export Date: 21 October 2024 Адрес для корреспонденции: Nezhinskij, V.M.; St. Petersburg State UniversityRussian Federation; эл. почта: v.nezhinskij@spbu.ru Адрес для корреспонденции: Petrov, M.V.; Herzen Russian State Pedagogical UniversityRussian Federation; эл. почта: tkuik@mail.ru

PY - 2024/9/1

Y1 - 2024/9/1

N2 - 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.

AB - 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.

KW - diagram

KW - disk-band graph

KW - isotopy

KW - tangle

KW - transformer

UR - https://www.mendeley.com/catalogue/4051acaf-b8c5-3a8d-97a6-b057cfa0be45/

U2 - 10.1134/s1063454124700171

DO - 10.1134/s1063454124700171

M3 - статья

VL - 57

SP - 331

EP - 334

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -