Jones polynomials and classical conjectures in knot theory

The primeness is necessary in the last statement ofTheorem B, since the connected sum of two figure eight knots is alternating, but it has a minimal non-alternating projection. Note that the figure eight knot is amphicheiral. Theorems A and B follow easily from Theorems l-4 (stated below) which show strong connections between c(L) and the Jones polynomial Vr(t). Let d maxVL(t) and d,i,v~(t) denote the maximal and minimal degrees of V,(t), respectively, and span V,(t) = d,,, Vr(t) d,;,VL(t).