I'm writing a thesis on something that involves Skein relations in terms of a certain "Markov distance". The latter is not important for this matter, but what I don't get is a way to simply explain how Skein relations arise from say the smoothing of a curve, see for example cluster algebras from surfaces by Schiffler pages 34-36 theorem 4.1.
Smoothing a curve $\gamma$ at a self-crossing gives one of the Skein relations, the one for self-crossings: $x_{\gamma}=y_{34}x_{\gamma_{34}}+y_{56}x_{\gamma_{56}}$ where $\gamma_{34}$ and $\gamma_{56}$ are curves resulting from the smoothing of $\gamma$ at its self-crossing and $y_{34},y_{56}$ are elements from something called a tropical semifield. (I would share related Figures I made, but I can't add them here somehow.) Now these $x$'s with subscript curves/multicurves are cluster variables, but what are they. Their alternative definition is in theorem 3.4, where $\gamma$ is an arc. What I have trouble understanding is what the meaning of this cluster variable, $x_{\gamma_{34}}$ for example, is. What values or meanings can this variable be? What is the intuition behind such a cluster variable?
Also, when are these elements $y_{ij}$ equal to $1$? I have only come across Skein relation used in theorems that have these $y$'s all $1$. Skein relations for smoothing a crossing point of two curves look a lot like Ptolemy relations, which also don't have these variables $y_{ij}$.
I have no experience with knot theory at all, but I feel that could maybe be related to this somehow, because I usually see Skein relations on the internet in the knot theory context. These relations are way older, but cluster algebras are relatively newer (2002).