徐妍,雷逢春,李风玲,梁良.有亏格为1的Heegaard分解的三维流形中的环面纽结[J].数学年刊A辑,2024,(1):1~14
有亏格为1的Heegaard分解的三维流形中的环面纽结
On Torus Knots in 3-Manifolds with Genus One Heegaard Splitting
Received:July 06, 2023  Revised:December 21, 2023
DOI:10.16205/j.cnki.cama.2024.0001
中文关键词:  H’-分解  透镜空间  环面纽结  Seifert流形  
英文关键词:H′-Splitting  Lens space  Torus knot  Seifert manifold
基金项目:国家自然科学基金(No.12071051)
Author NameAffiliation
XU Yan School of Mathematical Sciences, Dalian University of Technology,Dalian 116024, Liaoning, China. 
LEI Fengchun School of Mathematical Sciences, Dalian University of Technology,Dalian 116024, Liaoning, China. 
LI Fengling School of Mathematical Sciences, Dalian University of Technology,Dalian 116024, Liaoning, China. 
LIANG Liang School of Mathematics, Liaoning Normal University, Dalian 116029, Liaoning, China. 
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中文摘要:
      将存在亏格为1的Heegaard分解T’1∪F T’2的三维流形记为M=L(p,q),其中p和q是互素整数,q/p为T’2的纬线在T’1上的斜率.若环面F上的简单闭曲线γ在M中非平凡,则称γ是M中的环面纽结.本文对在M中沿环面纽结作m/n-Dehn手术所得流形进行了分类,并给出了两个实心环体沿边界上平环作融合所得流形是L(p,q)中环面纽结补的特征描述.
英文摘要:
      Let $M=\mathcal{L}(p,q)$ be a 3-manifold which admits a Heegaard splitting $T_1'\cup_F T_2'$ of genus 1, where $p$ and $q$ are co-prime integers, and a meridian curve of $T_2'$ has the slope $s=q/p$ on $T_1'$. A simple closed curve $\gamma$ on the torus $F$ is called a torus knot in $M$ if it is non-trivial in $M$. The main results of the paper are as follows: the authors classify the manifolds obtained by performing a $m/n$-Dehn surgery along a torus knot in $M$, and describe the characteristics for the manifold obtained by gluing two solid tori together along an annulus on the boundary of each solid torus to be a torus knot complement in $\mathcal{L}(p,q)$.
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