A Polytopal View of Classical Polynomials

结理论是对结和连杆的数学研究。结是一根两端绑在一起的缠结的绳子；一个链环由几个结缠在一起组成。结理论在自然科学中有广泛的应用，例如在DNA的研究中。结理论的一个基本难题是如何判断两个连接是否不同：一个连接能否在不解开绳子两端的情况下变形成另一个连接？将多项式与链接关联是解决这个问题的一种方法。本计画的目的是研究结理论和其他经典数学分支中的多项式，并将其与多面体联系起来。多面体是具有平坦边的任意尺寸的几何对象。对三维多面体的研究可以追溯到古代。该项目还包括对研究生的指导，以及对初高中学生的推广。多项式的支撑是它的单项式的指数向量的集合，这些单项式的系数是非零的。多项式的牛顿多面体是包含其支撑点的最小整数多面体。如果一个多项式的牛顿多面体上的每一个整数点都在它的支撑点上，那么它就是一个饱和的牛顿多面体。这些概念延伸到除一项基以外的其他基。本项目的目标是：(1)研究经典多元多项式的饱和性质与各种基，如单项式和舒伯特基；(2)它们所产生的整数多面体的研究；(3)对突出猜想的应用。这种方法的一个说明性的例子是Hafner， Mészáros和Vidinas在1962年Fox猜想的基础上的最新进展，该猜想指出，交替连杆的Alexander多项式系数的绝对值形成一个梯形序列。亚历山大多项式有许多组合模型，可以用来定义组合多元亚历山大多项式。对于其中一个模型，一个特殊交替连杆的关联组合多元亚历山大多项式的支持是广义复面体上整数点的集合。这样的多边形结果，与Brändén和Huh提出的洛伦兹多项式理论一起，证明了在特殊交变连杆情况下原始亚历山大多项式的对数凹性，从而证明了其梯形性质。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Knot theory is the mathematical study of knots and links. A knot is a single tangled string with the ends tied; a link consists of several knots tangled together. Knot theory has wide applications in the natural sciences, such as in the study of DNA. A basic difficult question of knot theory is how to tell if two links are different: can one be deformed to the other without untying the ends of the strings? Associating polynomials to links is one way to tackle this problem. The aim of this project is to study polynomials in knot theory and other classical branches of mathematics by associating polytopes to them. Polytopes are geometric objects in arbitrary dimensions with flat sides. The study of 3-dimensional polytopes dates back to ancient times. The project also involves mentoring of graduate students as well as outreach to middle and high school students.The support of a polynomial is the set of exponent vectors of its monomials appearing with nonzero coefficients. The Newton polytope of a polynomial is the smallest integer polytope containing its support. A polynomial has a saturated Newton polytope if every integer point in its Newton polytope is in its support. These notions extend to other bases besides the monomial basis. The goals of this project are (1) the study of saturation properties of classical multivariate polynomials with respect to various bases, such as the monomial and Schubert bases; (2) the study of the integer polytopes they give rise to; and (3) their applications to outstanding conjectures. An illustrative example of this approach is the recent progress by Hafner, Mészáros and Vidinas on Fox’s conjecture from 1962, which states that the absolute values of the coefficients of the Alexander polynomial of an alternating link form a trapezoidal sequence. There are many combinatorial models for the Alexander polynomial which can be used to define combinatorial multivariate Alexander polynomials. For one such model, the support of an associated combinatorial multivariate Alexander polynomial of a special alternating link is the set of integer points in a generalized permutahedron. Such polytopal results, together with the theory of Lorentzian polynomials developed by Brändén and Huh, enabled the proof of log-concavity, and thus trapezoidal property, of the original Alexander polynomial in the case of special alternating links.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.