The Jones Polynomial from a Goeritz Matrix

Joseph Boninger (CUNY GC)

Abstract: The Jones polynomial, defined in 1984, is a link invariant with deep applications in quantum physics and three-dimensional topology. It is also mysterious: it is an open problem, posed by Atiyah, to give a three-dimensional interpretation of the polynomial. We will briefly introduce the Jones polynomial, then share an original construction of the polynomial using a Goeritz matrix. Goeritz matrices are combinatorial objects, associated to link diagrams, that have been studied for almost ninety years. Our work also incorporates matroid theory, and has applications to links in thickened surfaces.