Ahmed Helali

• In this thesis, we review knot theory, focusing on its relationship with graph theory. We explore knots and their equivalence with signed planar graphs, explore Reidemeister moves on knots and their counterpart on signed planar graphs, and introduce the rank polynomial as a building block for a knot invariant w.r.t signed planar graphs. We then explore the Kauffman bracket and its relation with the rank polynomial. We then develop the Jones Polynomial utilizing both the rank polynomial and the Kauffman bracket. We show that the Jones polynomial is invariant under all Reidemeister moves w.r.t. both knots and signed planar graphs. In the end, we provide a working example on the right-handed trefoil.