Bounds in simple hexagonal lattice and classification of 11-stick knots

IF 0.3 4区数学 Q4 MATHEMATICS

Journal of Knot Theory and Its Ramifications Pub Date : 2024-02-27 DOI:10.1142/s0218216523500979

Yueheng Bao, Ari Benveniste, Marion Campisi, Nicholas Cazet, Ansel Goh, Jiantong Liu, Ethan Sherman

引用次数: 0

Abstract

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial $11$ -stick knots in the sh-lattice are the trefoil knot ( $3_{1}$ ) and the figure-eight knot ( $4_{1}$ ).

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简单六方格中的界限和 11 棍结的分类

简单六方格（sh-lattice）中的绳结类型的棍数和边长分别是在 sh-lattice 中构建给定类型的绳结所需的最小棍数和边长。通过引入网格间的线性变换，我们证明了对于任何给定的结，sh-网格中的两个值都严格小于立方网格中的值。最后，我们证明了在 sh 格中唯一的非难 11 棍结是三叶草结 (31) 和八字结 (41)。

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

Journal of Knot Theory and Its Ramifications 数学-数学

CiteScore

0.80

自引率

40.00%

发文量

127

审稿时长

4-8 weeks

期刊介绍： This Journal is intended as a forum for new developments in knot theory, particularly developments that create connections between knot theory and other aspects of mathematics and natural science. Our stance is interdisciplinary due to the nature of the subject. Knot theory as a core mathematical discipline is subject to many forms of generalization (virtual knots and links, higher-dimensional knots, knots and links in other manifolds, non-spherical knots, recursive systems analogous to knotting). Knots live in a wider mathematical framework (classification of three and higher dimensional manifolds, statistical mechanics and quantum theory, quantum groups, combinatorics of Gauss codes, combinatorics, algorithms and computational complexity, category theory and categorification of topological and algebraic structures, algebraic topology, topological quantum field theories). Papers that will be published include: -new research in the theory of knots and links, and their applications; -new research in related fields; -tutorial and review papers. With this Journal, we hope to serve well researchers in knot theory and related areas of topology, researchers using knot theory in their work, and scientists interested in becoming informed about current work in the theory of knots and its ramifications.