We give lower bounds for the tunnel number of knots and handlebody-knots. We also give a lower bound for the cutting number, which is a “dual” notion to the tunnel number in handlebody-knot theory. We provide necessary conditions to be constituent handlebody-knots by using G-family of quandles colorings. The above two evaluations are obtained as the corollaries. Furthermore, we construct handlebody-knots with arbitrary tunnel number and cutting number.

中文翻译：

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通道号和切割号以及构成的车体结

我们给结和句柄结的隧道数量下界。我们还给出了切割数量的下限，这是手柄体打结理论中隧道数量的“双重”概念。我们通过使用quand着色的G系列来提供必要的条件来构成手柄体结。以上两个评估是推论。此外，我们构造具有任意隧道数和切割数的手柄体结。