# Singular crossings and Ozsv\'ath-Szab\'o's Kauffman-states functor

## TL;DR

This paper extends Ozsváth and Szabó's algebraic framework for knot Floer homology to include singular crossings, using holomorphic disk counts in bordered sutured Heegaard diagrams, advancing the algebraic tools for knot invariants.

## Contribution

It introduces bimodules for singular crossings that generalize existing structures, based on holomorphic disk counting in bordered sutured diagrams.

## Key findings

- Defined bimodules for singular crossings.
- Connected bimodule construction to holomorphic disk counts.
- Enhanced algebraic tools for knot Floer homology.

## Abstract

Recently, Ozsv\'ath and Szab\'o introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras B(n) and, for a generator of the braid group on n strands, a certain type of bimodule over B(n). We define analogous bimodules for singular crossings. Our bimodules are motivated by counting holomorphic disks in a bordered sutured version of a Heegaard diagram considered previously by Ozsv\'ath, Stipsicz, and Szab\'o.