A Polytopal View of Classical Polynomials

纽结理论是对纽结和链接的数学研究。一个结是一个单一的纠缠字符串的两端绑;一个链接由几个结纠缠在一起。纽结理论在自然科学中有着广泛的应用，例如DNA的研究。纽结理论的一个基本难题是如何判断两个链环是否不同：一个链环是否可以变形为另一个链环而不解开绳子的末端？将多项式与链接相关联是解决这个问题的一种方法。这个项目的目的是研究多项式在纽结理论和其他经典分支的数学相关联的多面体。多面体是具有平坦边的任意维度的几何对象。三维多面体的研究可以追溯到古代。该项目还包括指导研究生以及向初中和高中学生推广。多项式的支集是其单项式的指数向量的集合，其系数为非零。多项式的牛顿多面体是包含其支撑的最小整数多面体。一个多项式有一个饱和的牛顿多面体，如果它的牛顿多面体中的每个整数点都在它的支撑中。这些概念扩展到除了单项式基之外的其他基。这个项目的目标是：（1）研究经典多元多项式关于各种基的饱和性质，如单项式和Schubert基;（2）研究它们产生的整数多面体;（3）它们在杰出代数中的应用。这种方法的一个说明性的例子是哈夫纳，Mészáros和Vidinas对福克斯猜想从1962年的最新进展，其中指出，交替链接的亚历山大多项式的系数的绝对值形成梯形序列。亚历山大多项式的组合模型有很多种，它们可以用来定义组合多元亚历山大多项式。对于一个这样的模型，特殊交替链接的相关组合多元亚历山大多项式的支持是广义置换面体中的整数点集。这种多面体的结果，加上Bränden和Huh发展的洛伦兹多项式理论，使得在特殊的交替链接的情况下，证明了原始亚历山大多项式的对数周期，从而梯形属性。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.