A formula for the braid index of links

Morton and Franks–Williams independently gave a lower bound for the braid index b ( L ) of a link L in S 3 in terms of the v-span of the Homfly-pt polynomial P L ( v , z ) of L: 1 2 span v P L ( v , z ) + 1 ⩽ b ( L ) . Up to now, many classes of knots and links satisfying the equality of this Morton–Franks–Williamsʹs inequality have been founded. In this paper, we give a new such a class K of knots and links and make an explicit formula for determining the braid index of knots and links that belong to the class K . This gives simultaneously a new class of knots and links satisfying the Jones conjecture which says that the algebraic crossing number in a minimal braid representation is a link invariant. We also give an algorithm to find a minimal braid representative for a given knot or link in K .