 Knots and Their Polynomials The value of the Jones polynomial does not depend on the orientation chosen Let us focus on one crossing in an oriented knot diagram: If we reverse the orientation and rotate the diagram 180 degrees, that crossing appears exactly as it did before. So reversing the orientation in a diagram and then applying the skein relation to a crossing is the same as rotating the original diagram 180 degrees, applying the skein relation, rotating the products of the relation another 180 degrees, and then reversing the orientation. Symbolically, letting ``'' represent the orientation change, Rot a 180degree rotation, and Sk the effect of applying the skein relation to the knot diagram K, we have Sk ( K) =  (Rot Sk Rot K) where we apply operations, as usual, from right to left. Suppose our analysis involves two applications of the skein relation Sk_{2} Sk_{1} ( K) = Sk_{2} ( Rot Sk_{1} Rot K) =  (Rot Sk_{2} Rot Rot Sk_{1} Rot K) =  (Rot Sk_{2} Sk_{1} Rot K), since two consecutive 180degree rotations of the entire figure have the same effect as no rotation at all, and similarly for more than two applications. Suppose that it takes n applications of the skein relation to reduce all the diagrams to unknots, for which the Jones polynomial (symbolically J{ }) does not depend on orientation. Then J{( K)} = J{Sk_{n} ... Sk_{1} ( K)} = J{  (Rot Sk_{n} ... Sk_{1} Rot K)} = J{Rot Sk_{n} ... Sk_{1} Rot K} (since the Jones polynomial of an unknot diagram does not depend on the orientation) = J{Sk_{n} ... Sk_{1} K} = J{K} (since a rotated knot is topologically the same). So the Jones polynomial of the reversedorientation diagram is the same as the Jones polynomial of the original diagram. So why is an orientation required? Because to apply the skein relation correctly at a crossing, we need the two strands to be coherently oriented. Otherwise the calculation derails, as a little experimentation will show. On to the next knot page. Back to the previous knot page. Back to the first knot page.

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