Quantum group and its application to knot polynomial was the theme of Witten-Reshtikhin-Turaev theory.

The categorification of quantum group and application to knot homology has been discovered and developed by Khovanov-Lauda, Rouquier, and Webster, in the form of KLRW algebra. With Mina Aganagic and collaborators, we developed a new approach using Fukaya category and relate to KLRW algebra. The geometric origin allows us to give a geometric construction for the categorified coproduct. The talk is based on past work and work in progress with Mina Aganagic, Vivek Shende, Elise LePage, Ivan Danilenko and Yixuan Li.