The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov’s novel construction of “the categorification of the Jones polynomial”. For the same low cost we also provide some computations, including one that shows that Khovanov’s invariant is strictly stronger than the Jones polynomial and including a table of the values of Khovanov’s invariant for all prime knots with up to 11 crossings.
"On Khovanov's categorification of the Jones polynomial." Algebr. Geom. Topol. 2 (1) 337 - 370, 2002. https://doi.org/10.2140/agt.2002.2.337