有亏格为1的Heegaard分解的三维流形中的环面纽结

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.

On Torus Knots in 3-Manifolds with Genus One Heegaard Splitting

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 $\mathcal{L}(p,q)$.