Tools of the trade

It's winter break, so here's a short post.  I am trying to spend some time doing new non-math things.  The other day I was trying to help out with a fence repair.  If you have spent time with me, you'd likely think that I would not know how to help someone repair a fence - if you thought that, you are indeed correct.  I was not particularly helpful, but while I was embarrassing myself not knowing how to use a hammer, I did gain some insight into the mathematical process, which I wanted to mention here.

In low-dimensional topology, as I hope to convey in this blog, it is very beneficial to have a wide range of tricks and tools.  This includes knowing topological tools (e.g. the Montesinos trick or Kirby calculus), invariants (e.g. gauge theory, Khovanov homology), but also perspectives and connections from other fields (e.g. viewing torus knots as links of singularities of algebraic curves).  However, knowing that hammers exist is not the same as being able to use a hammer.  Upon quick reminder, it was pointed out to me that the back part of a hammer can be used for pulling out nails - seems simple enough.  But after significant strain with no gain, it was clear that: a) I was doing something very wrong and b) everyone else there could have done this with no problem.  Someone handed me a small scrap of wood to wedge against the hammer for leverage, and then magically the nail came out!  I thought this person was a genius, until I became aware that this is standard fare in hammerology.  If you do hammering (or I guess inverse-hammering), then this seems to be a standard trick.  In math, even if you've seen the tools, it can seem like the experts have this amazing insight into the tools that a n00b could never imagine.  But, I think a lot of this mystery is actually just experience in disguise.  The more you know how to use a tool, the easier it is to apply in standard settings, but also the easier it is to use it in original settings as well.  This is why I think it's so useful to talk to as many people as you can and to look at lots of papers - you can learn how the experts wield the tools.  It's also important to practice.  When you learn a new tool, try it out on your favorite open problems, use it to reprove known results, to build conversations with other people, etc.  Also, try to make it your own, so you can find your own perspectives and applications of it as well.