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JOURNALS // Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy // Archive

Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2024 Volume 11, Issue 3, Pages 489–494 (Mi vspua310)

MATHEMATICS

Disk-band graphs in the theory of framed tangles

V. M. Nezhinskiia, M. Petrovb

a Saint Petersburg State University
b The Herzen State Pedagogical University of Russia, 48, nab. r. Moyki, St. Petersburg, 191186, Russian Federation

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.

UDC: 515.162.8, 519.17

MSC: 57M15

Received: 28.01.2024
Revised: 15.02.2024
Accepted: 22.02.2024

DOI: 10.21638/spbu01.2024.305



© Steklov Math. Inst. of RAS, 2025