We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov homology or HOMFYLPT homology from there? When googling, one can find a lot of papers containing certain categorifications of algebras but I find it hard to pinpoint which of these arise most naturally regarding my question.
Much of the research in knot homology has been about categorifying these algebras!
Khovanov's paper math/0103190 is devoted to defining and studying the Temperley-Lieb 2-category which is a categorification of the Temperley-Lieb category. (The TL algebras arise and endomorphisms of objects in the TL category.)
Later, many authors have studied the category of Soergel bimodules, which is a categorification of the Hecke algebra. This approach was used by Khovanov-Rozansky math/0510265 to define HOMFLYPT homology.