Quandle colorings are invariant under oriented Reidemeister moves

An R1 kink adds one underarc color $y$ to the unchanged boundary arc color $x$. Depending on the local orientation, the crossing equation is $y=x\ast x$ or $x=y\ast x$. In the first case idempotency gives the unique extension $y=x$; in the second case $x=x\ast x$ and right invertibility give the unique solution $y=x$. Removing the kink is the inverse bijection.

At an R2 move, fix the color $y$ on the strand passing over the other strand. The understrand's color changes by $R_y$ at one crossing and by $R_y^{-1}$ at the other; these maps are inverse bijections by [F1]. Thus every boundary coloring extends uniquely across the two-crossing side, and the inverse R2 move gives the inverse map.

For the braid-like R3 orientation, the local crossing map is $B(x,y)=(y,x\ast y)$ and its inverse is . Starting from $(x,y,z)$, the two sides of R3 give respectively $(z,y\ast z,(x\ast y)\ast z)$ and $(z,y\ast z,(x\ast z)\ast (y\ast z))$; these are equal by right self-distributivity. For every $z$, $R_z(x\ast y)=(x\ast y)\ast z=(x\ast z)\ast (y\ast z)$, so $R_z$ preserves $\ast$. If , then $u\ast b=a$; applying the bijection $R_z$ gives $R_z(u)\ast R_z(b)=R_z(a)$, and uniqueness of right division yields . Hence $R_z$ preserves $\bar{\ast }$ too, and its inverse preserves both operations. Thus for either operation ${\star }_1,{\star }_2\in \{\ast ,\bar{\ast }\}$, $(x{\star }_{1}y){\star }_{2}z=(x{\star }_{2}z){\star }_1(y{\star }_{2}z)$. These four identities give the R3 relation for every choice of crossing direction, and the bijective crossing maps give the corresponding extension bijections. A planar isotopy simply transports each arc label.

For the converse, the R1 extension property gives $x\ast x=x$ for every $x$: on the kink side, the boundary color $x$ can extend only when this equality holds.

In an R2 tangle with over-color $y$, the under-color map is $R_y$. Unique extension for every incoming color makes $R_y$ a function defined everywhere; unique extension in the reverse direction means each target color has exactly one preimage. Hence $R_y$ is bijective for every $y$.

In the braid-like R3 tangle, arbitrary boundary colors $x,y,z$ yield the outputs $(x\ast y)\ast z$ and $(x\ast z)\ast (y\ast z)$. Unique extension to the same boundary coloring forces these outputs to agree for all triples, which is right self-distributivity.

Thus the local axioms are necessary and sufficient for the stated local extension property. Under ${\mathrm{A}\mathrm{C}}_{\omega }$, [F3] connects diagrams of the same oriented link by finitely many such moves; composing the local bijections gives a bijection of coloring sets. A finite quandle has finitely many assignments on a finite diagram, so the bijection preserves their counts. Countable choice is used only through [F3]; the move-by-move calculations in steps 1.1–1.6 are choice-free.