TITLE: How to motivate the skein relations?
QUESTION [13 upvotes]: I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this skein relation to compute all these wonderful polynomial invariants of knots. 
I was not trained as a knot theorist, so I was at loss. Intuitively  it does not seem to be that powerful, because it is not clear (to me at least) that you can always pick a recursive relation involving simpler knots. Could somebody help me motivating my students here? In other words,
Why are the skein relations so useful for computing polynomial invariants of knots, and when did people realize that is the case? 
Thanks.

REPLY [13 votes]: One of the earliest appearances of the ingredients for a skein relation can be found in Romilly Allen's 1904 book on Celtic Knotting. There he explains that designers of Celtic knot patterns first start with a regular weave and then systematically replace some of the crossings by "horizontal" and "vertical" smoothings. These days, it is easy to find working demonstrations of this process on the internet. The skein triple consists in an unoriented crossing and its two possible smoothings. This is the skein triple underlying the bracket polynomial model for the Jones polynomial and directly related to Temperley-Lieb algebra. Of course neither Romilly Allen nor the Celtic knotters had any intention to calculate polynomial invariants of knots! That came much later. Another important point about this simplest skein relation is that it is related (by shading the link diagram) to the contraction and deletion operations on the edges of a plane graph. This is why the bracket polynomial and the Jones polynomial are related to the Dichromatic (Tutte) polynomial and to signed generalizations of the Tutte polynomial. And then finally, the two smoothings of a crossing are related to one another in that they form the 'top' and 'bottom' of a saddle surface that joins them. This is the key geometric idea behind Khovanov homology.

REPLY [3 votes]: Here is a sketch of how the skein relations appear in the approach to knot invariants based on braided monoidal categories coming e.g. from representations of quantum groups. 
Suppose $V$ is a dualizable object in a braided monoidal category $(C, \otimes)$. This means, among other things, that $V^{\otimes n}$ acquires an action of the braid group $B_n$, and moreover we can meaningfully take traces of the action of a braid in a way which produces an invariant of the knot obtained by closing the braid up. Famously, the Jones polynomial arises in this way when $C$ is taken to be the braided monoidal category of representations of the quantum group $U_q(\mathfrak{sl}_2)$ and $V$ is taken to be the standard $2$-dimensional representation. 
Now, one way to describe the skein relation satisfied by the Jones polynomial is that it is a linear relation between three endomorphisms of $V^{\otimes 2}$, namely

two parallel strands, describing the identity,
a crossing, describing the braiding $b_{V, V} : V^{\otimes 2} \to V^{\otimes 2}$, and
the other crossing, describing the inverse $b_{V, V}^{-1}$.  

Now, why might there be a linear relation between these three elements? Certainly a sufficient condition is if $\text{End}(V^{\otimes 2})$ is at most $2$-dimensional. And this actually happens in the Jones polynomial case. (The point is that the representation theory of $U_q(\mathfrak{sl}_2)$ is sufficiently similar to that of $\mathfrak{sl}_2$ that it remains true that $V^{\otimes 2}$ is a direct sum of two nonisomorphic irreducibles.)