Algebra by pictures

Abstract: 

Maybe you've heard of diagram algebras, planar algebras, "spiders" or something similar. And you're wondering, is this really math? Are the pictures an analogy or are they actual mathematical objects? Happily, there are rigorous mathematical frameworks that have pictures as their ingredients. I'll tell you about some of the most exciting examples, including the Temperley-Lieb algebra (and its relation to knot theory), the color-counting planar algebra (and the five-color theorem), and linear algebra with pictures.