We show that the problem of determining whether a knot in the 3-sphere is non-trivial lies in NP. This is a consequence of the following more general result. The problem of determining whether the Thurston norm of a second homology class in a compact orientable irreducible 3-manifold with (possibly empty) toroidal boundary is equal to a given integer is in NP. As a corollary, the problem of determining the genus of a knot in the 3-sphere is in NP.

中文翻译：

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打结和瑟斯顿范数的有效证明

我们表明，确定 3 球体中的结是否重要的​​问题在于 NP。这是以下更一般结果的结果。确定具有（可能为空）环形边界的紧致可定向不可约 3-流形中第二个同源类的 Thurston 范数是否等于给定整数的问题在 NP 中。作为推论，确定 3 球体中结的属的问题是 NP。