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In this paper I shall show how certain bounds on the possible diagrams presenting a given oriented knot or link K can be found from its two-variable polynomial P K defined in  . The inequalities regarding exponent sum and braid index of possible representations of K by a closed braid which are proved in  and  follow as a special case. Notation. In a diagram D for an oriented knot, write c + (D) and c~(D) for the number \ / of positive and negative crossings, where V is a positive crossing.doi:10.1017/s0305004100063982 fatcat:z3tbebpqujbytipl22up42bc6y