Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 112 / 2

Apply the Kauffman bracket skein relation to every crossing in the two-string tangle $D_1$. Each complete smoothing is a collection of closed circles together with one of the two crossingless pairings of the four boundary points. Rotation through $180^{\circ }$ about the indicated vertical axis preserves both crossingless pairings and the number of closed circles. Gluing the unchanged tangle $D_2$ to corresponding smoothings therefore gives equal terms, including equal loop factors, so $⟨D⟩=⟨D^{\prime }⟩$.

Choose an orientation of $L$, and transport the induced orientations of the four ends of $D_1$ through the rotation to orient $L^{\prime }$. Crossing signs inside the rotated tangle and outside it then have the same total, so the writhe of a link diagram satisfies $w(D)=w(D^{\prime })$. Applying the writhe normalization of the Jones polynomial gives

The component knot types can change under this Conway mutation. To construct an example, place a knotted arc carrying a trefoil knot summand in one strand of $D_1$ and a knotted arc carrying a figure-eight knot summand in one strand of $D_2$, leaving the other two strand portions trivial. Choose the outside pairing so that before rotation the two knotted portions lie on different components, while after rotation they lie on the same component. Then the component multisets arerespectively. A link isotopy preserves the unordered multiset of component knot types, so these links are not isotopic, although part (a) shows that suitable orientations give them the same Jones polynomial.