A graphical approach to the Melvin-Morton Conjecture, Part I

Our aim is to give a proof of the Melvin-Morton Conjecture—the “truncated” Jones polynomial is equal to the reciprocal of the Alexander polynomial—in terms of techniques from knot and graph theory. In this paper, we show that the Melvin-Morton Conjecture holds for 3-braids. The techniques developed in this paper form a basis for a proof of the Melvin-Morton Conjecture for all braids, which we will discuss in a subsequent paper, Part II.