We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. We show that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to 4-ball genus. We also show a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times.

翻译：我们引入了一种新的实际价值的变异性,称为三层中双曲结的自然斜坡,其定义是其角形几何。我们显示,结的签字和自然斜坡是结的两倍,而自然斜坡的两倍,最多是由投射半径的立方体除以的超曲体体体体体积相隔的经常时间。这种不平等是利用机器学习来探测不同结结的成形体之间的关系而发现的。它适用于Dehn手术和4球基因。我们还展示了一种精确的不平等,即上界是体积的线性函数,而斜坡则用与短的大地学相匹配的术语加以纠正,该词将结点与奇数联系起来。