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.
Best Answer
Regarding "when", it was Alexander, in his paper on what we call the Alexander polynomial. Conway was the first to popularize them, I believe.
Why are they useful? I'm not sure I believe they are so useful. Sometimes I'm interested in computing Alexander invariants but the knots and links that I'm looking at do not have easy-to-compute diagrams associated to them. Say you have a homology 3-sphere and you apply the JSJ-decomposition to it. This produces some knots and links in homology spheres but working out diagrams can be a pain. So sometimes it's far easier to use the covering space definition to get at the Alexander polynomials. Moreover, Skein relations don't give you the Alexander module and how Poincare duality works on the module, or naturality (when you have maps between 3-manifolds), etc.