**Data**

- Title: -webs, categorification and Khovanov-Rozansky homologies
- Author: Daniel Tubbenhauer
- Status: To appear in J. Knot Theory Ramifications. Last update: Fri, 2 Oct 2020 05:17:24 EST
- ArXiv link: http://arxiv.org/abs/1404.5752
- LaTex Beamer presentation: Slides1, Slides2, Slides3

**Abstract**

**A few extra words**

It turns out that every -web can be obtained by a string of only 's acting as elements of on a highest weight vector. Thus, we can say that the -web space is an instance of -highest weight theory.

A closed -web is therefore nothing else than a quantum number since one jumps from the -dimensional highest weight space to the -dimensional lowest weight space using the 's of .

As a consequence: One can write each (colored) link as string of 's together with operators for crossings. As for example displayed below for the Hopf link (with restriction to ). Here we act with the 's from bottom to top and the -weight spaces can be read of from the number grid.

The operators measure the difference how one can hop around from the highest to the lowest weight and we obtain the Reshetikhin-Turaev -link polynomials therefore as an instance of -highest weight theory.

History repeats itself: The same is true for the -link homologies. We just replace the weight modules with weight categories, the 's with functors and the measurement of the difference with certain natural transformations.

And since everything an the level of the -link polynomials takes place in a certain -module of highest weight we can obtain the Khovanov-Rozansky -link homologies by only using the cyclotomic KL-R algebra.

Thus, these homologies are instances of categorified -highest weight theory.