A Polytopal View of Classical Polynomials

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.