Representations of branched twist spins with a non-trivial center of order 2

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-02-18 DOI:10.1016/j.topol.2025.109284

Mizuki Fukuda

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引用次数: 0

Abstract

A branched twist spin is a 2-knot consisting of exceptional orbits and fixed points of a circle action on the four sphere. It is a generalization of the twist spun knot, and its knot group is obtained from a Wirtinger presentation of the original 1-knot group by adding a generator corresponding to a regular orbit of the circle action and a certain relator. In this paper, we study on

{SL}_{2} (Z_{3})

-representations and dihedral group representations. For the former case, we give a sufficient condition for the existence of an

{SL}_{2} (Z_{3})

-representation for a branched twist spin. For the latter case, we determine the number of 4k-ordered dihedral group representations of branched twist spins. As an application, we can show non-equivalence between two branched twist spins by counting dihedral representations of their knot groups.

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具有二阶非平凡中心的支旋自旋的表示

支旋自旋是由特殊轨道和在四个球体上作用的圆的固定点组成的2结。它是捻旋结的推广，它的结群是在原1-结群的Wirtinger表示的基础上，通过增加一个对应于圆作用的规则轨道的产生子和一定的关联子而得到的。本文研究了SL2(Z3)-表示和二面体群表示。对于前一种情况，我们给出了支旋自旋的SL2(Z3)-表示存在的充分条件。对于后一种情况，我们确定了支旋自旋的4k有序二面体群表示的数量。作为一个应用，我们可以通过计算两个分支旋的结群的二面体表示来证明它们之间的不等价。

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.