Planar equivalence of knotoids and quandle invariants

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-05-06 DOI:10.1016/j.topol.2025.109407

Mohamed Elhamdadi , Wout Moltmaker , Masahico Saito

引用次数: 0

Abstract

While knotoids on the sphere are well-understood by a variety of invariants, knotoids on the plane have proven more subtle to classify due to their multitude over knotoids on the sphere and a lack of invariants that detect a diagram's planar nature. In this paper, we investigate equivalence of planar knotoids using quandle colorings and cocycle invariants. These quandle invariants are able to detect planarity by considering quandle colorings that are restricted at distinguished points in the diagram, namely the endpoints and the point-at-infinity. After defining these invariants we consider their applications to symmetry properties of planar knotoids such as invertibility and chirality. Furthermore we introduce an invariant called the triangular quandle cocycle invariant and show that it is a stronger invariant than the end specified quandle colorings.

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结点和纠缠不变量的平面等价

虽然球体上的结样可以通过各种不变量来很好地理解，但平面上的结样被证明更难以分类，因为它们在球体上有大量的结样，并且缺乏检测图的平面性质的不变量。本文研究了平面类结的等价性，并给出了双色和环不变量。这些纠缠不变量能够通过考虑纠缠着色来检测平面性，这些着色被限制在图中的不同点，即端点和无穷远处的点。在定义了这些不变量之后，我们考虑了它们在平面类结的可逆性和手性等对称性质上的应用。在此基础上，我们引入了一个称为三角纠缠循环不变量的不变量，并证明了它是一个比末端指定纠缠着色更强的不变量。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.