The Möbius energy is one of the knot energies, and is named after its Möbius invariant property. It is known to have several different expressions. One is in terms of the cosine of conformal angle, and is called the cosine formula. Another is the decomposition into Möbius invariant parts, called the decomposed Möbius energies. Hence the cosine formula is the sum of the decomposed energies. This raises a question. Can each of the decomposed energies be estimated by the cosine formula? Here we give an affirmative answer: the upper and lower bounds, and modulus of continuity of decomposed parts can be evaluated in terms of the cosine formula. In addition, we provide estimates of the difference in decomposed energies between the two curves in terms of Möbius invariant quantities.

莫比乌斯能量是结能量之一，并以其莫比乌斯不变性命名。已知有几种不同的表达方式。一种是关于保角角的余弦，称为余弦公式。另一个是分解为莫比乌斯不变部分，称为分解的莫比乌斯能量。因此，余弦公式是分解的能量之和。这就提出了一个问题。可以通过余弦公式估算每个分解的能量吗？在这里，我们给出一个肯定的答案：可以根据余弦公式评估分解部分的上限和下限以及连续性模量。此外，我们提供了根据莫比乌斯（Möbius）不变量的两条曲线之间的分解能差异的估计。