Diagrammatic representations of 3-periodic entanglements

IF 0.6 4区 数学 Q3 MATHEMATICS
Toky Andriamanalina , Myfanwy E. Evans , Sonia Mahmoudi
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引用次数: 0

Abstract

Diagrams enable the use of various algebraic and geometric tools for analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures (TP tangles), which are embeddings of simple curves in R3 that are invariant under translations along three non-coplanar axes. As such, these entanglements can be seen as preimages of links embedded in the 3-torus T3=S1×S1×S1 in its universal cover R3, where two non-isotopic links in T3 may possess the same TP tangle preimage. We consider the equivalence of TP tangles in R3 through the use of diagrams representing links in T3. These diagrams require additional moves beyond the classical Reidemeister moves, which we define and show that they preserve ambient isotopies of links in T3. The final definition of a tridiagram of a link in T3 allows us to then consider additional notions of equivalence relating non-isotopic links in T3 that possess the same TP tangle preimage.
三周期纠缠的图解表示
图表可以使用各种代数和几何工具来分析和分类结。本文介绍了三周期纠缠结构(TP缠结)的一种新的图解表示,它是在R3中沿三个非共面轴平移时不变的简单曲线的嵌入。因此,这些缠结可以被视为嵌入在其通用覆盖层R3中的3环面T3=S1×S1×S1中的链接的预像,其中T3中的两个非同位素链接可能具有相同的TP缠结预像。我们通过使用表示T3中的链路的图来考虑R3中TP缠结的等价性。这些图需要在经典的Reidemeister移动之外进行额外的移动,我们定义并表明它们保留了T3中链接的环境同位素。T3中链接的三角图的最终定义允许我们考虑与具有相同TP缠结原像的T3中非同位素链接相关的其他等效概念。
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
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.
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