AI 中文总结
该研究定义了交错链的一个四变量Laurent多项式不变量，推广了Alexander多项式的生成树公式，证明其相关序列满足梯形性质，并提出了关于该多项式的对称性等猜想。
AI 中文摘要
我们定义了交错链的一个不变量——齐次四变量Laurent多项式，它编码了对称化Alexander多项式、符号及其他拓扑数据。在此过程中，我们将Murasugi和Stoimenow提出的Alexander多项式的生成树公式从特殊交错链推广到所有交错链。该研究受Fox的梯形猜想驱动，因此我们证明，对于所有交错链，与该不变量相关的某些序列是梯形的。我们还猜想该多项式具有M-凸支撑，且满足对称性和对数凹性，并证明了部分对称性结果。
英文摘要
We define an invariant of alternating links---a homogeneous, four-variable Laurent polynomial---that encodes the symmetrized Alexander polynomial, the signature, and other topological data. Along the way, we extend a spanning tree formulation of the Alexander polynomial due to Murasugi and Stoimenow from special alternating links to all alternating links. This project is motivated by Fox's trapezoidal conjecture; accordingly, we prove certain sequences associated to our invariant are trapezoidal for all alternating links. We also conjecture our polynomial has $M$-convex support, and that it satisfies symmetry and log-concavity properties. We prove a partial symmetry result.
Comments39 pages, many figures, comments welcome
