论文标题
对结预测和广义时钟定理的离散视角
A discrete Morse perspective on knot projections and a generalised clock theorem
论文作者
论文摘要
我们获得了一个简单而完整的表征，即在结图的泰特图上匹配，在$ s^2 $上诱导离散的摩尔斯函数（DMF），从而扩大了由于科恩而延伸的结构。我们显示这些DMF在泰特图中的某些生根跨越森林中进行了两者。我们使用它来计算涉及图拉普拉斯式的封闭公式的此类DMF的数量。然后，我们同时概括了考夫曼的时钟定理和肯尼恩·普罗普·威尔森的信函，沿两个不同的方向进行了概括。我们首先证明对应关系的图像在完美的DMF上诱导了两次射击，然后我们证明所有符合可接纳性条件的完美匹配都与有限的点击和时钟移动序列相关。最后，我们研究和比较了与塔特图相关的匹配和离散的摩尔斯音乐络合物，以部分kauffman的态度，并提供一些计算。
We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on $S^2$, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.
