Alexander Bjurström: The Correspondence Between the Tutte and Jones Polynomial with Applications

Abstract: This paper explores the relationship between the Tutte polynomial of graphs and the Jones polynomial of knots. We begin by introducing the necessary background from graph theory and knot theory, including properties of the Tutte and Jones polynomials. We then show that the Jones polynomial of an alternating knot can be obtained as a specialization of the Tutte polynomial of the associated Tait graph. In the second part of the paper, we apply this correspondence to compute the Jones polynomial of alternating knots in several knot families: twist knots, pretzel knots, torus knots and rational knots.