TY - JOUR

T1 - Meander diagrams of knots and spatial graphs

T2 - Proofs of generalized Jablan-Radovic conjectures

AU - Belousov, Yury

AU - Malyutin, Andrei

N1 - Funding Information: The reported study was funded by RFBR according to the research project n. 17-01-00128 A. Publisher Copyright: © 2020 Elsevier B.V. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - We study decomposition into simple arcs (i.e., arcs without self-intersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph G is a knotted loop, then there exists a plane diagram D of G such that (i) each edge of G is represented by a simple arc of D and (ii) each vertex of G is represented by a point on the boundary of the convex hull of D. This generalizes the conjecture of S. Jablan and L. Radovic stating that each knot has a meander diagram, i.e., a diagram composed of two simple arcs whose common endpoints lie on the boundary of the convex hull of the diagram. Also, we prove another conjecture of Jablan and Radovic stating that each 2-bridge knot has a semi-meander minimal diagram, i.e., a minimal diagram composed of two simple arcs. (C) 2020 Elsevier B.V. All rights reserved.

AB - We study decomposition into simple arcs (i.e., arcs without self-intersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph G is a knotted loop, then there exists a plane diagram D of G such that (i) each edge of G is represented by a simple arc of D and (ii) each vertex of G is represented by a point on the boundary of the convex hull of D. This generalizes the conjecture of S. Jablan and L. Radovic stating that each knot has a meander diagram, i.e., a diagram composed of two simple arcs whose common endpoints lie on the boundary of the convex hull of the diagram. Also, we prove another conjecture of Jablan and Radovic stating that each 2-bridge knot has a semi-meander minimal diagram, i.e., a minimal diagram composed of two simple arcs. (C) 2020 Elsevier B.V. All rights reserved.

KW - Knot

KW - Meander

KW - Simple arc

KW - Spatial graph

UR - http://www.scopus.com/inward/record.url?scp=85080126367&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2020.107122

DO - 10.1016/j.topol.2020.107122

M3 - Article

AN - SCOPUS:85080126367

VL - 274

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

M1 - 107122

ER -