TITLE: Links which HOMFLY homology distinguish but the HOMFLY polynomial does not.
QUESTION [10 upvotes]: Does anyone know of a pair of different links which the HOMFLY polynomial does not distinguish, but HOMFLY homology does? Or does there exist such a pair of links? 

I'm assuming there does exist such a pair, but have never seen it. I've been looking for this for a few days and have had no luck finding or computing such an example. Any help would be greatly appreciated!

REPLY [11 votes]: Although $5_1$ and $10_{132}$ cannot be distinguished by Jones, Alexander and (uncolored) HOMFLY-PT polynomials, their HOMFLY homologies do tell them. (See the review by Gukov-Saberi.)
In addition, some mutant pairs can be distinguished by Khovanov homology. (See the paper by Wehrli.)