On a proof of the Labastida-Marino-Ooguri-Vafa conjecture

We outline a proof of a remarkable conjecture of Labastida-Mari{\~n}o-Ooguri-Vafa about certain new algebraic structures of quantum link invariants and the integrality of infinite family of new topological invariants. Our method is based on the cut-and-join analysis and a special rational ring characterizing the structure of the Chern-Simons partition function.