We explore the relationship between two knot invariants, unknotting number and crosscap number. Specifically, we present two infinite families of knots with a specific crossing change that lowers the unknotting number while raising the crosscap number. One of these families is hyperbolic. This is a non-orientable parallel of Scharlemann and Thompson’s 1988 result for orientable surfaces.

中文翻译：

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提高crosscap数同时降低unknotting数

我们探讨了两个结不变量，解结数和交叉帽数之间的关系。具体来说，我们提出了两个无限系列的结，它们具有特定的交叉变化，降低了未打结的数量，同时提高了交叉帽的数量。这些家族之一是双曲线的。这是 Scharlemann 和 Thompson 1988 年可定向表面结果的不可定向平行线。