Jones grading from symplectic Khovanov homology

Symplectic Khovanov homology is first defined by Seidel and Smith as a singly graded link homology. It is proved isomorphic to combinatorial Khovanov homology over any characteristic zero field by Abouzaid and Smith. In this dissertation, we construct a second grading on symplectic Khovanov homology from counting holomorphic disks in a partially compactified space. One of the main theorems asserts that this grading is well-defined. We also conclude the other main theorem that this second grading recovers the Jones grading of Khovanov homology over any characteristic zero field, through showing that the Abouzaid and Smith's isomorphism can be refined as an isomorphism between doubly graded groups. The proof of the theorem is carried out by showing that there exists a long exact sequence in symplectic Khovanov homology that commutes with its combinatorial counterpart