标题： 关于纽结的瞬态数 显示英文标题

标题： On the transient number of a knot

摘要： 一个扭结K的瞬态数，记为tr(K)，是必须附加到K上的简单弧的最小数目，以便K可以在K和这些弧的并集的正则邻域内被同伦为一个平凡扭结。 我们给出了tr(K)的一个下界，该下界涉及K的二重分支覆盖空间的一阶同调群的秩。特别是，如果t(K)=1，则K的二重分支覆盖空间的一阶同调群是循环的。 利用这一点，我们可以计算表格中许多扭结的瞬态数，并证明存在瞬态数任意大的扭结。 显示英文摘要

摘要： The transient number of a knot K, denoted tr(K), is the minimal number of simple arcs that have to be attached to K, in order that K can be homotoped to a trivial knot in a regular neighborhood of the union of K and the arcs. We give a lower bound for tr(K) in terms of the rank of the first homology group of the double branched cover of K. In particular, if t(K)=1, then the first homology group of the double branched cover of K is cyclic. Using this, we can calculate the transient number of many knots in the tables and show that there are knots with arbitrarily large transient number.