Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of \(S^3\) branched along a knot \(\alpha \subset S^3\). Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot \(\alpha \) can be derived from dihedral covers of \(\alpha \). The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications.

令M是一个连通的、闭合的、有向的三流形，而K、  L是M中的两条有理零同调有向简单闭合曲线 。我们给出了一个明确的算法，用于计算K和L之间的链接数，将 M表示为沿结\(\alpha \subset S^3分支的\(S^3\)的不规则二面三倍覆盖 \)。由于每个封闭的、定向的三流形都承认这样的表示，因此我们的结果适用于所有三流形中的所有（明确定义的）连接数。此外，结 \(\alpha \)的带状障碍可以从 \(\alpha \)。我们计算的连接数对于评估一种这样的障碍物是必要的。这项工作是朝着测试切片丝带猜想的潜在反例以及其他应用迈出的一步。