结和热带曲线
Knots and tropical curves
摘要：
利用热带几何中的基本思想，我们给每个有理函数的$q$ -完整序列赋予一条热带曲线。特别是，我们分配一个热带曲线，这是由琼斯多项式的结和它的平行线确定的每一个结。主题曲线解释了AJ猜想和斜率猜想之间的关系（将纽结的琼斯多项式及其平行线与$\SL(2,\BC)$特征变量和不可压缩曲面的斜率联系起来）。我们的讨论预测，热带曲线是对偶的牛顿细分的$A$ -多项式的结。我们明确地计算了$4_1$、$5_2$和$6_1$结的热带曲线，并验证了上述预测。
英文摘要：
Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Conjecture and the Slope Conjecture (which relate the Jones polynomial of a knot and its parallels to the $\SL(2,\BC)$ character variety and to slopes of incompressible surfaces). Our discussion predicts that the tropical curve is dual to a Newton subdivision of the $A$-polynomial of the knot. We compute explicitly the tropical curve for the $4_1$, $5_2$ and $6_1$ knots and verify the above prediction.
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