SURGERY n-TRIVIALITY AND COMPANION TORI

We define a knot K to be surgery n-trivial if there exists a collection of m-disjoint crossing circles for K such that twisting along any 0<m≤n of them turns K into the unknot. We use techniques from the theory of sutured 3-manifolds to study the question of whether, in the case that K is a non-trivial satellite, these crossing circles can be chosen to be disjoint from a companion torus of K. We show that if the twists are taken to have order bigger than one the answer to this question is yes. As a consequence of this we show that, under the hypothesis that the twists used have order bigger than one, the only knot that is surgery n-trivial for all n∈N, is the unknot.

我们定义一个纽结\(K\)为手术\(n\)-平凡的，如果存在\(K\)的\(m\)个不相交的交叉圆的集合，使得沿着其中任意\(0 < m\leq n\)个交叉圆进行扭转能将\(K\)变为平凡纽结。我们使用缝合\(3\)-流形理论的技术来研究当\(K\)是一个非平凡的卫星纽结时，这些交叉圆是否可以被选取为与\(K\)的伴随环面不相交的问题。我们表明如果扭转被假定为阶大于\(1\)，那么这个问题的答案是肯定的。作为这个结论的一个结果，我们表明，在所用扭转的阶大于\(1\)的假设下，对于所有\(n\in N\)都是手术\(n\)-平凡的唯一纽结是平凡纽结。