The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that twisted, generalized Whitehead doubles of a knot satisfies the Slope Conjecture and the Strong Slope Conjecture if the original knot does. Additionally, we provide a proof that there are Whitehead doubles which are not adequate.

中文翻译：

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扭曲广义怀特黑德双打的强斜率猜想

Garoufalidis 提出的斜率猜想断言，有色琼斯多项式的阶数决定了边界斜率，它的细化，Kalfagianni 和 Tran 提出的强斜率猜想断言，阶数中的线性项决定了满足以下条件的基本曲面的拓扑结构斜率猜想。在某些假设下，我们证明，如果原始结满足斜率猜想和强斜率猜想，则结的扭曲的广义怀特黑德对偶满足。此外，我们提供了一个证据，证明怀特黑德双打是不够的。