# Bordered Heegaard Floer homology

### bfh_python

In a sequence of papers, Peter Ozsváth, Dylan Thurston, and I gave an algorithm for computing the Heegaard Floer invariant HF-hat of a closed 3-manifold and the Ozsváth-Szabó spectral sequence associated to a link in the 3-sphere. We also gave a computer implementation of the algorithm for computing HF-hat. In his thesis, Bohua Zhan improved the algorithm, using a notion of "local DA bimodules". He also re-implemented the original and improved algorithms, and extended the implementation to comptue the Ozsváth-Szabó spectral sequence.

Zhan's code is bfh_python, available here. If you are interested in using bordered Heegaard Floer homology for computations, you should use this.

I have written some documentation introducing bfh_python, available here. This was supported by NSF grant DMS-1810893.

### Archaic code referenced in my papers

The following code is largely archaic; use bfh_python instead.

- A technology demonstration of a package to compute Heegaard Floer invariants using bordered Floer homology is available here. This is referenced in "Computing HF-hat by factoring mapping classes".
- You will need Sage to run this. I have not checked that it still works in current versions of Sage.
- This is version 0.2 (August 5, 2011). It corrects several (critical) bugs found by Bohua Zhan.
- This version contains new installation options, thanks to Nathan Dunfield. See the README (also courtesy of Dunfield).
- Partial documentation is included, in a PDF called BordProgDocs
- Copyright 2010-2012 Robert Lipshitz, Peter Ozsvath and Dylan Thurston.
- This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
- This was supported by NSF Grant DMS-0905796 and a Sloan Research Fellowship.

- An extension of part of the bordered Floer package was used to perform computations for "Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers" is available here.
- This is version 1 (March 14, 2012).
- Partial documentation is included.
- The extension is Copyright 2012 Robert Lipshitz and David Treumann.
- This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
- This was supported by NSF Grant DMS-1149800.

# Khovanov homology

Sucharit Sarkar and I constructed a stable homotopy refinement of Khovanov homology and gave a formula for computing the induced map Sq^{2} on Khovanov homology. We also used this to give a refinement of Rasmussen's s-invariant. We (mostly Sucharit) implemented these algorithms in code. These were supported by NSF Grant DMS-1149800.

- Code for computing the actions of Sq
^{1}and Sq^{2}on Khovanov homology. Used in our paper "A Steenrod square on Khovanov homology". - Code for computing the refined s-invariant from Sq
^{2}on Khovanov homology. Used in and based on our paper "A refinement of Rasmussen's s-invariant".

There is also faster code for doing these kinds of computations, written by others:

- Dirk Schütz program KnotJob and its variants.
- Cotton Seed's program KnotKit.

*Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.*