An improvement of the lower bound for the minimum number of link colorings by quandles

IF 0.3 4区数学 Q4 MATHEMATICS

Journal of Knot Theory and Its Ramifications Pub Date : 2022-10-12 DOI:10.1142/s021821652350044x

H. Abchir, Soukaina Lamsifer

引用次数: 0

Abstract

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548, 2018). We express this lower bound in terms of the degree k of the reduced Alexander polynomial of the considered knot. We show that it is exactly k + 1 for L-space knots. Then we apply these results to torus knots and Pretzel knots P(-2,3,2l + 1), l>=0. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more that one component.

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一种改进的用纠缠进行最小链路着色数下界的方法

我们改进了Fox:the other linear Alexander quandles（linear Algebra and its Applications，Vol.5481018）的着色定理1.2中给出的结的线性Alexander quandle着色的最小颜色数的下界。我们用所考虑的结的约化亚历山大多项式的次数k来表示这个下界。我们证明，对于L空间节点，它恰好是k+1。然后我们将这些结果应用于环面结和椒盐脆饼结P（-2,3,2l+1），l>=0。我们注意到，对于某些特定的结，可以达到这个下限。此外，我们证明了上面引用的定理1.2可以扩展到具有多个分量的链接。

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.