THE JONES POLYNOMIAL AND DESSINS D’ENFANT OLIVER T. DASBACH, DAVID FUTER, EFSTRATIA KALFAGIANNI, XIAO-SONG LIN, AND NEAL W. STOLTZFUS Abstract. The Jones polynomial of an alternating link is a certain specializa- tion of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollob´ as–Riordan–Tutte polynomial gener- alizes the Tutte plolynomial of planar graphs to graphs that are embedded in closed surfaces of higher genus (i.e. dessins d’enfant). In this paper we show that the Jones polynomial of any link can be obtained from the Bollob´ as–Riordan–Tutte polynomial of a certain dessin associated to a link projection. We give some applications of this approach. 1. Introduction Dessins d’enfant were introduced by Grothendieck and proved to be useful in many areas of mathematics (see e.g. [Sch94, LZ04] for introductions). Informally, dessins are graphs with a cyclic orientation around their vertices. Their genus is the mini- mal genus of an orientable surface in which the dessin embeds. Recently, Bollob´ as and Riordan extended the Tutte polynomial to dessins [BR01] as a three-variable polynomial, where the third variable is related to the genus of dessins. The dessins of interest in this introductory paper arise naturally from link pro- jections. We will show that the Jones polynomial, via the Kauffman bracket, is a specialization of the Bollob´ as-Riordan-Tutte polynomial of these dessins. Further- more, this approach leads to a natural dessin-genus of a knot as the minimal genus of the dessin of a knot diagram over all knot projections. Knots of dessin-genus 0 are exactly the alternating knots. Our approach is different from the one taken by Thistlethwaite [Thi87], where he gives a spanning tree expansion of the Jones polynomial via signed graphs. Using these ideas Kauffman defined a three-variable Tutte polynomial for signed graphs Date : May 19, 2006. The first author was supported in part by NSF grants DMS-0306774 and DMS-0456275 (FRG). The second author was supported in part by NSF grant DMS-0353717 (RTG). The third author was supported in part by NSF grants DMS-0306995 and DMS-0456155 (FRG). The fourth author was supported in part by NSF grants DMS-0404511 and DMS-0456217 (FRG). The fifth author was supported in part by NSF grant DMS-0456275 (FRG). 1