用3色琼斯多项式预测纽结补集体积，准确率达99.34%。
Colored Jones Polynomials and the Volume Conjecture
- 通过顶点模型计算超双曲纽结的3色琼斯多项式
- 神经网络从多项式预测体积，准确率99.34%
- 提出更优相位猜测，改进体积猜想收敛性
利用辫表示的顶点模型方法，计算了15个交叉以下超双曲纽结上自旋-1对应的多项式，称为3色琼斯多项式或伴随琼斯多项式。使用全连接前馈神经网络对部分数据进行训练，可从伴随琼斯多项式或其在特定相位 $q=e^{8πi/15}$ 处的取值中，以99.34%的准确率预测纽结补集的体积。分析2色与3色琼斯多项式后，我们提出了 $n$-色琼斯多项式最优相位的猜想，并据此改进体积猜想的表述。该改进在已知 $n$-色琼斯多项式闭式表达的纽结上得到验证，显示体积逼近效果显著提升。
原文摘要 · Abstract (English)
Using the vertex model approach for braid representations, we compute polynomials for spin-1 placed on hyperbolic knots up to 15 crossings. These polynomials are referred to as 3-colored Jones polynomials or adjoint Jones polynomials. Training a subset of the data using a fully connected feedforward neural network, we predict the volume of the knot complement of hyperbolic knots from the adjoint Jones polynomial or its evaluations with 99.34% accuracy. A function of the adjoint Jones polynomial evaluated at the phase $q=e^{ 8 πi / 15 }$ predicts the volume with nearly the same accuracy as the neural network. From an analysis of 2-colored and 3-colored Jones polynomials, we conjecture the best phase for $n$-colored Jones polynomials, and use this hypothesis to motivate an improved statement of the volume conjecture. This is tested for knots for which closed form expressions for the $n$-colored Jones polynomial are known, and we show improved convergence to the volume.
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