For p ≥ 1, one can define a generalisation of the unknotting number tup called the pth untwisting number, which counts the number of null-homologous twists on at most 2p strands required to convert the knot to the unknot. We show that for any p ≥ 2 the difference between the consecutive untwisting numbers tup–1 and tup can be arbitrarily large. We also show that torus knots exhibit arbitrarily large gaps between tu1 and tu2.

中文翻译：

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连续解捻数之间的间隙

为了p≥ 1，可以定义解结数的一般化涂p叫做pth untwisting number，最多统计2个null-homlogous twists的个数p将结转换为未结所需的股线。我们证明，对于任何p≥2 连续解捻数之差涂p–1和涂p可以任意大。我们还表明，圆环结之间存在任意大的间隙涂1和涂2.