Pseudoholomorphic quilts and Khovanov homology

TL;DR

This paper explores the symplectic Khovanov homology, establishing its structures, relations to Khovanov homology, and defining symplectic analogues of Khovanov's arc algebras, advancing the geometric understanding of knot invariants.

Contribution

It introduces symplectic analogues of Khovanov's arc algebras and demonstrates their isomorphism with the original algebras, linking symplectic and combinatorial knot invariants.

Findings

Symplectic Khovanov homology has structures analogous to Khovanov homology.

Defined symplectic arc algebras $H_{s}^m$ and proved their isomorphism to Khovanov's $H^m$.

Established a skein exact triangle for symplectic Khovanov homology.

Abstract

We further study the symplectic Khovanov homology of Seidel and Smith and its generalization to even tangles. This homology theory is a conjectural geometric model for Khovanov homology. In this paper we uncover structures on symplectic Khovanov homology which have analogues in Khovanov homology. To each elementary (as well as minimal) cobordism between two tangles we associate a homomorphism between the symplectic Khovanov homology groups of the two tangles. We define the symplectic analogues $H_{s}^m$ of Khovanov's arc algebras and equip the symplectic Khovanov homology of an $(m,n)$-tangle with the structure of an $(H_{s}^m,H_{s}^n)$-bimodule. We show that $H_{s}^m$ and Khovanov's $H^m$ are isomorphic as associative algebras over $\Z/2$. We also obtain a skein exact triangle for symplectic Khovanov homology which resembles the one for Khovanov homology.