Two Applications of the ■■-Hodge Theory

Using Hodge theory and Banach fixed point theorem,Liu and Zhu developed a global method to deal with various problems in deformation theory.In this note,the authors generalize Liu-Zhu's method to treat two deformation problems for non-Kahler manifolds.They apply the ■■-Hodge theory to construct a deformation formula for(p,q)-forms of compact complex manifold under deformations,which can be used to study the Hodge number of complex manifold under deformations.In the second part of this note,by using the ■■-Hodge theory,they provide a simple proof of the unobstructed deformation theorem for the non-Kahler Calabi-Yau ■■-manifolds.

New structures for colored HOMFLY-PT invariants

In this paper,we present several new structures for the colored HOMFLY-PT(Hoste-Ocneanu-Millet-Freyd-Lickorish-Yetter-Przytycki-Traczyk)invariants of framed links.First,we prove the strong integrality property for the normalized colored HOMFLY-PT invariants by purely using the HOMFLY-PT skein theory developed by Morton and his collaborators.By this strong integrality property,we immediately obtain several symmetric properties for the full colored HOMFLY-PT invariants of links.Then we apply our results to refine the mathematical structures appearing in the Labastida-Mari?o-Ooguri-Vafa(LMOV)integrality conjecture for framed links.As another application of the strong integrality,we obtain that the q=1 and a=1 specializations of the normalized colored HOMFLY-PT invariant are well-defined link polynomials.We find that a conjectural formula for the colored Alexander polynomial which is the a=1 specialization of the normalized colored HOMFLY-PT invariant implies that a special case of the LMOV conjecture for framed knots holds.