Gukov et al. suggested triply graded link homology groups via refined BPS counting on the deformed conifold. Through large N duality they identify their Poincar\'e polynomials for the Hopf link as refined topological vertices. I further apply the geometric engineering to interpret them as holomorphic Euler characteristics of natural vector bundles over Hilbert schemes of points on the affine plane. Then they perfectly make sense mathematically. This work is very preliminary, but I hope it could be developed further. Most of time, I will give expository explanation of backgrounds, which I hope to be understandable to non-experts, and then finally state my result.

nakajima@math.kyoto-u.ac.jp