Consider a rigid braided monoidal category, with braiding $\beta_{x,y} : x \otimes y \cong y \otimes x$, and every object has a dual such that $\epsilon_x : 1 \to a \otimes a^*, \bar\epsilon_x : a^* \otimes a \to 1$ satisfying the zig-zag identities. Now we have the Reidemeister moves, e.g. $$ (c \otimes \beta_{a,b}) \circ \beta_{a\otimes b, c} = \beta_{b \otimes a, c} \circ (\beta_{a,b}\otimes c) $$ saying that braiding the two string with another, and then braiding the two, is the same as first braiding the two and then with the other. This is Reidemeister III. Similarly $\beta_{a\otimes a^*, b} \circ (b \otimes \epsilon_a) = \epsilon_a \otimes b$ is Reidemeister II, and $(\bar\epsilon_{a} \otimes a) \circ (a^* \otimes \beta_{a,a}) \circ (\epsilon_a\otimes a) = \mathrm{id}_a$ is Reidemeister I. This is rather like the Morse link presentation of knots and links.

Is there any reference that develops such an idea? How can the theory of quandles be integrated in this picture? I'm guessing that quandles either behave like modules over a knot category, or we have a joint generalization of quandles and knot categories.