{"title":"Representations of branched twist spins with a non-trivial center of order 2","authors":"Mizuki Fukuda","doi":"10.1016/j.topol.2025.109284","DOIUrl":null,"url":null,"abstract":"<div><div>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 <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>-representations and dihedral group representations. For the former case, we give a sufficient condition for the existence of an <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>-representation for a branched twist spin. For the latter case, we determine the number of 4<em>k</em>-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.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"365 ","pages":"Article 109284"},"PeriodicalIF":0.6000,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topology and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166864125000823","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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 -representations and dihedral group representations. For the former case, we give a sufficient condition for the existence of an -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.
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.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
