RUI: Link Homology Theories and Other Quantum Invariants

California State University-Fresno Foundation, Fresno CA

Investigators

Abstract

Low-dimensional topology is a branch of mathematics that studies spaces of low dimensions, including knots, singular links, and virtual knots, as well as knotted surfaces. Invariants of knots and links are efficient tools that allow us to distinguish between such mathematical objects. Study of link invariants has yielded powerful new invariants in the form of homology theories that arise through categorification, the process of replacing a known invariant for links with a family of algebraic objects that generalize and enrich the original invariant. One focus of the project is to better understand some existing link homologies and investigate their properties and applications, as well as to construct new link homologies. Another goal of the project is to extend known quantum invariants to other knot-like objects, including singular links. The award will support research experiences in which students are charged with investigating and working on publication-worthy open questions. The PI is committed to increasing the participation of women in mathematics and will also continue to organize the university's annual Sonia Kovalevsky Math Day, an event designed for students in grades 7-12, with the goal of empowering the next generation of female mathematicians, scientists, engineers, and innovators. Moreover, the PI organizes a series of talks aimed for college students, celebrating the achievements of young mathematicians, in particular women and underrepresented groups in mathematical sciences; speakers are encouraged to talk not only about math and their mathematical achievements but also about their path and endeavors to a career in mathematical sciences. The range of topics in this project will establish connections between various areas of mathematics, including low-dimensional topology, combinatorics, abstract algebra, and representation theory. The project aims to construct and study new Khovanov-type homology theories for classical knots and singular knots via webs and foams modulo relations and to investigate applications of these theories to questions about braids and cobordisms, in particular to questions related to concordance, ribbon distance, and invariants of surface-links and surface-knots. Another goal of the project is to extend classical quantum invariants to singular links and virtual links by means of skein modules, Markov traces, and Birman-Murakami-Wenzl algebras, as well as combinatorial techniques involving graphical calculus. There are manageable problems for students stemming from the project. 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.