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Based on large N Chern-Simons/topological string duality, in a series of papers [36, 21, 19], Labastida, Marino, Ooguri, and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.

We introduce a random recursive method for constructing random Salem sets in$R$^{d}. The method is inspired by Salem's construction [13] of certain signular monotonic functions.

No abstract uploaded!

No abstract uploaded!