Kauffman bracket curl and writhe correction

Label the crossing ports northwest $1$, northeast $2$, southeast $3$, and southwest $4$, in the chart of [F1]. The over-strand joins $4$ to $2$ and the under-strand joins $1$ to $3$. For the left exterior cap $(1,4)$ the remaining ports are $2,3$. Orient its through strand from $2$ to $3$: it traverses the over-strand northeast to southwest, then the cap, then the under-strand northwest to southeast. The tangent vectors may be taken as $(-1,-1)$ and $(1,-1)$, whose determinant is $2>0$. This is a positive kink. The A-pairing $(1,4)(2,3)$ closes one additional circle against the cap; the B-pairing $(1,2)(3,4)$ gives just the through arc. Hence its bracket factor is $A\delta +A^{-1}=-A^3$. Reversing the component negates both tangent vectors and preserves the determinant.

For the top exterior cap $(1,2)$ the remaining ports are $4,3$. Orient its through strand from $4$ to $3$: it traverses the over-strand southwest to northeast, then the cap, then the under-strand northwest to southeast. The tangent vectors $(1,1)$ and $(1,-1)$ have determinant $-2<0$, so this is a negative kink. Here the A-pairing gives only the through arc, while the B-pairing closes one additional circle. Its factor is $A+A^{-1}\delta =-A^{-3}$. The right and bottom cap cases are the respective $180$-degree rotations; orientation-preserving chart changes and reversal of the component preserve the calculation. These are all oriented type-I variants. The calculation is inside the move disk and permits every exterior connectivity and any additional components. Deletion uses the reciprocal Laurent-unit factor.

The positive kink changes writhe from $w$ to $w+1$, and the negative kink changes it to $w-1$. Their respective normalized values are $(-A^3{)}^{-(w+1)}(-A^3)⟨D⟩=(-A^3{)}^{-w}⟨D⟩,$ and $(-A^3{)}^{-(w-1)}(-A^{-3})⟨D⟩=(-A^3{)}^{-w}⟨D⟩\mathrm{.}$ Thus both type-I additions and their inverse deletions preserve $f_D$.

A type-II move creates or deletes two opposite signs, and a type-III move preserves all three signs, so writhe is unchanged. Their raw bracket is unchanged by [F4]. Planar isotopy transports the determinant crossing signs, smoothing states and state-circle counts bijectively. Therefore it preserves both factors of $f_D$, and the normalized value is invariant under every listed move.