Unknotting nonorientable surfaces of genus 4 and 5

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-08-22 DOI:10.1016/j.laa.2024.08.014

Mark Pencovitch

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引用次数: 0

Abstract

Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in $D^{4}$ with knot group $Z_{2}$ .

In particular we show that if two such surfaces have the same normal Euler number, the same fixed knot boundary K such that $| \det (K) | = 1$ , and the same nonorientable genus 4 or 5, then they are ambiently isotopic rel. boundary.

This implies that closed, nonorientable, locally flatly embedded surfaces in the 4-sphere with knot group $Z_{2}$ of nonorientable genus 4 and 5 are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that an obstruction to modified surgery is elementary. Furthermore we show that this method fails for nonorientable genus 6 and 7.

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解结属 4 和 5 的非定向曲面

在康威、奥森和鲍威尔工作的基础上，我们研究了结群为 Z2 的 D4 中不可定向、紧凑、局部平嵌曲面的同位类相对边界。特别是，我们证明了如果两个这样的曲面具有相同的法欧拉数、相同的固定结边界 K（使得 |det(K)|=1 ）以及相同的不可定向属 4 或 5，那么它们就是环境同位类相对边界。这意味着在 4 球中，具有不可定向的结群 Z2 的不可定向属 4 和 5 的封闭、不可定向、局部平嵌曲面在拓扑上是无结的。证明依赖于在 Sage 中实现的计算，这意味着修正手术的障碍是基本的。此外，我们还证明了这种方法对于不可定向的属 6 和属 7 不适用。

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

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.