Functoriality and boundary-collar action for the three-manifold skein quotient

Define $\tilde{f}:{\mathcal{F}}_R(M)\to {\mathcal{F}}_R(N)$ on basis elements by $\tilde{f}([L])=[f(L)]$ and extend $R$-linearly. A diffeomorphism sends ambient-isotopic framed annuli to ambient-isotopic framed annuli and sends the framing annuli by its derivative, so this is well-defined on the free modules.

Fix framed-link representatives $P$ in $\mathrm{\Sigma }\times (0,1)$ and $L$ in the interior of $M$. To check independence of isotopy representatives, take any isotopy track of either framed link. Its framing annuli have compact trace in the interior. Place the trace of the product-factor link in a sufficiently thin outer collar layer disjoint from the trace of the $M$-factor link. The disjoint-union family is a smooth isotopy of framed annuli in $M$. In finitely many submanifold charts of its compact trace, extend the annulus velocity to ambient charts with cutoffs and combine these extensions with a partition of unity; the resulting compactly supported time-dependent vector field has a flow that realizes this isotopy by ambient diffeomorphisms. Thus the union class is independent of both representatives. For the quotient relations, rescale $P$ into a boundary layer disjoint from $L$. A change of $P$ by either defining relation becomes the identical local relation after the collar embedding by [F2]; a change of $L$ by either relation remains the identical relation with the disjoint collar link outside its ball. The same local check in two successive collar layers shows that stacking descends to the skein quotients. Hence collar gluing gives a bilinear action. Decreasing the collar width uses an orientation-preserving product rescaling isotopy supported in a thin boundary layer and disjoint from $L$, so the resulting skein class is unchanged.

The image of an oriented crossing ball is an oriented ball. Its transported chart preserves the over/under crossing and both endpoint pairings, so $\tilde{f}([L_{\times }]-A[L_A]-A^{-1}[L_B])$ is the corresponding defining relation in $N$. A zero-framed trivial circle bounding a disk in the ball maps to another zero-framed trivial circle bounding the image disk. Thus $\tilde{f}({\mathcal{R}}_A(M))\subseteq {\mathcal{R}}_A(N)$ and $\tilde{f}$ descends to $f_{\ast }$. Applying the same construction to $f^{-1}$ gives the inverse; applying it to a composite gives the composite map on every link generator.

Place two algebra links in successive subcollars and the $M$-link deeper inside. Both $(P\ast Q)\cdot L$ and $P\cdot (Q\cdot L)$ are represented by the same ordered union, up to an orientation-preserving product rescaling, and the empty link acts as the identity. Thus the action is associative and unital. If an orientation-preserving diffeomorphism carries one specified collar productwise to another, it sends each stacked or glued link union to the corresponding union, so its module maps intertwine the actions.