An oddification of Khovanov's arc algebras

TL;DR

This thesis introduces an oddification of Khovanov's arc algebras, resulting in non-associative rings that connect to the odd cohomology of Springer varieties, expanding algebraic structures in knot theory.

Contribution

It constructs a novel oddification of Khovanov's rings, creating non-associative algebraic structures linked to Springer variety cohomology.

Findings

Defined non-associative rings $OH^n_C$ with sign choices

Extended centers to include anti-commutative elements

Established isomorphism with odd cohomology of Springer varieties

Abstract

In this master thesis we construct an oddification of the rings $H^n$ from arXiv:math/0103190 using the functor from arXiv:0710.4300 . This leads to a collection of non-associative rings $OH^n_C$ where $C$ represent some choices of signs. Extending the center up to anti-commutative elements, we get a ring $OZ(OH^n_C)$ which is isomorphic to the oddification of the ring cohomology of the $(n,n)$-Springer variety from arXiv:1203.0797 .