A package to compute the Heergaard Floer correction terms for some classes of 3-manifolds. Information and examples concerning the correction terms may be found in a paper by P. Ozsvath, Z. Szabo, On the Heegaard Floer homology of branched double-covers.

Specifically, the package computes the correction terms for the following:

• Double branched cover of an alternating link (input is an alternating link)

  • Input type 1: Knotilus database
  • Input type 2: Plink drawing

• Plumbed 3-manifold constructed from a negative-definite weighted graph with with at most two bad vertices

  • Input type 1: Seifert fibered rational homology sphere data {e; (p1, q1),...,(pr, qr)}
  • Input type 2: Negative-definite graph with an most two bad vertices

This program was written for a Caltech SURF project summer 2013 with mentor Dr. Yi Ni. Funding was also provided by Richter Memorial Funds.

Installation the easy way requires pip first, with instructions here. If you do not have an installation of plink installed that python can find, run

pip install -f http://math.uic.edu/t3m/plink plink

before trying to install hfhom. The program source code is available on GitHub. To install this program, find the repository https://github.com/th0114nd/hfhom or https://github.com/panaviatornado/hfhom that is most up to date, then run either

[sudo] pip install git+git://github.com/th0114nd/hfhom.git
or
[sudo] pip install git+git://github.com/panaviatornado/hfhom.git

To test the installation, run

hfhom

For a successful installation, the program should start running.

If for some reason you cannot install it properly, you can download all the files in the directory corrterm and run gui.py with python. You will also have to install the necessary python modules, if you do not have them installed already.

To uninstall, run

[sudo] pip uninstall hfhom

It is recommended you run the GUI using a terminal or the command line, i.e. calling hfhom from the terminal. This will provide information about the program's current progress and intermediate results. On Linux, if the computation is taking too long or using too many resources, you can cancel the computation using Ctrl-C. You can also close the entire GUI if it is non-responsive via Ctrl-\ (Ctrl + backslash).

On the File menu, check the option for Use multiprocessing to use all the threads on your computer instead of just one. This will speed up lengthy computations and most likely max out your CPU usage (for knots/links with enough crossings). This is only useful if your computer has multiple cores or has multithreading enabled.

The GUI has two tabs, one for computing the corrections terms of the double branched cover of an alternating link, and one for computing the correction terms of a plumbed 3-manifold. Input for the double branched cover of an alternating link is an alternating link from either Knotilus or Plink. Input for a plumbed 3-manifold is either Seifert data for a Seifert fibered rational homology sphere or a negative definite weighted graph with at most two bad vertices.

From the inputted data, a negative definite quadratic form is computed, and then the Heegaard Floer correction terms are computed from the quadratic form.

The Knotilus archive section has two input methods, either by entering an archive number or opening a saved file download from the Knotilus database.

• Entering an archive number
  Archive numbers must be of the form ax-b-c, for integers a, b, and c. For example, 6x-1-1 or 20x-5-10. This method requires an Internet connection, and may take up to 20 seconds to load the link for 11 or more crossings. Visiting the link's database page with your browser and letting it load first will significantly decrease the program's running time. If the checkbutton Save file is selected, the plaintext data from the Knotilus database is saved in the current directory as ax-b-c.txt. Press the Go button when finished entering the archive number. See here for more details about the Knotilus archive number.

• Loading a downloaded Knotilus file
  A valid Knotilus file is created by going to [Knotilus] (http://knotilus.math.uwo.ca/), finding the desired link, then selecting Download > Plaintext and saving the file. The program will run noticeably faster on a downloaded Knotilus file than if it must fetch the file from the database. The option original link will only work if the filename is of the form ax-b-c.txt or ax-b-c. It will be ignored otherwise.

The PLink/SnapPy section has two input methods, either by drawing a new link using the PLink Editor or opening a saved PLink file. Instruction for using PLink can be found in the documentation for SnapPy, here.

• Drawing a new link
  Clicking the Create New button will open the PLink editor. Draw the link in the editor. Ensure the link is alternating, or use the menu option Tools > Make alternating. When finished drawing the link, close the window. A dialog to save the file will appear. If you choose not to save the file, the program will not be able to check that your link is closed and alternating, and the option original link will be ignored. If the link is not closed and alternating, results will be unpredictable.

• Loading a saved PLink file
  A valid PLink file is created by drawing a link in the PLink editor, then using the menu option File > Save .... Clicking the Open button will load the open file dialog to select a valid PLink file.

Data for a Seifert fibered rational homology sphere is represented as a list [e,(p1,q1),...,(pr,qr)], where e and all the pi, qi are integers, and all pi > 1 with gcd(pi, qi) = 1. If the resulting quadratic form is not negative definite, an error message will pop up.

This input method is for plumbed 3-manifolds constructed from a negative- definite weighted graph with at most 2 bad vertices.

Opening the editor will produce a "Graph controls" dialog.

Note: Increasing the size of the pyplot window will space out nodes more, which is useful in the case that they overlap.

• Creating a node
  A parent of -1 indicates that the node is a root node, i.e. has no parent. To create a node, select its desired parent node, enter the node's weight (integer), then click create. The graph will be draw using matplotlib.pyplot. Nodes are labeled Nk,w, where k is the node's number and w is its weight. For example, if the first node had weight -3, it would be labeled N0,-3.
• Editing a node
  To edit a node, select the node's number. The node's parent and weight will be edited at the same time. Enter the desired new data, then click Done. To leave the parent unchanged, select same from the dropdown menu. To leave the weight unchanged, leave the entry field empty.
• Deleting a node Only the last node (highest index) can be deleted.
• Drawing/Saving/loading/exiting
  • The Draw graph button will open pyplot and draw the current graph. The
  • program will automatically do this each time a node is created, edited, or deleted; this button is useful in case you close the pyplot window and don't want to create, edit, or delete a node or load a new graph.
  • Click the Save button to save the graph data in a plaintext file. The file will contain the adjacency list of the graph, followed by a list of the node attributes.
  • Click the Load button to load graph data from a plaintext file. The file must be in the same format used by Save. Loading a file will remove the current graph shown in pyplot. After loading a file, nodes may be created or edited. Note: The program uses the eval() method to parse the node attribute data. Thus, don't put Python code there that will harm your computer. (The program will raise an error if the data isn't a list, so doing this accidentally would be practically impossible.)
  • Click the Done/compute button to close the editor and compute the correction terms.
  • Click the Cancel button to close the editor without computing anything.
  • If attempting to compute the correction terms, an error message will pop up if the quadratic form is not negative definite.

Under the Options menu, there are three global options, Print quadratic form, Print H_1(Y) type', and Condense correction terms. Additionally, there are the submenus Double branched coverandPlumbed 3-manifolds`, which only affect those input methods, respectively. If an option starts with "Print", it will print additional information in the correction terms window. If an option starts with "Show", it will open a new window.

• Global Options
  If the quadratic form option is checked, the quadratic form (square matrix) will be printed, in addition to the correction terms, in the output window. Checking the condense correction terms box will disable the quadratic form, graph commands, and Seifert data options, and the output window will just contain the Knotilus archive number or filename, followed by a space and the correction terms, all on a single line. For the quadratic form option with Seifert data, if the manifold orientation is reversed, the quadratic form will be for the altered (reversed) manifold, rather than the original. The option `Print H_1(Y) type' is checked by default; it prints the group structure of the first homology group H_1(Y).

• Double branched cover submenu
  If the Show original link option is checked, a separate window will open to show the original link diagram. For Knotilus, this will open an Internet browser tab to the appropriate link. It may not work on Windows. If opening a saved Knotilus file, this option will only succeed if the filename is of the form ax-b-c.txt or ax-b-c, where 'ax-b-c' is the archive number. For PLink or SnapPy, the PLink editor will open with the original link drawing. The PLink file must be saved in order to do this.

• Plumbed 3-manifold submenu
  If the Show weighted graph option is checked, the graph will be drawn in a separate window using matplotlib.pyplot. The option Print modified Seifert data only affects Seifert data input.

This is the main command line interface, used for double branched cover and Seifert data.

Usage:

$ python corrterms.py [-k] [-m] archive_num
$ python corrterms.py -kf [-m] archive_num.txt
$ python corrterms.py -p [-m] [filename]
$ python corrterms.py -s [-m] '[e, (p1, q1),...,(pr, qr)]'

Command line options:

[-k] to download and save Knotilus archive_num plaintext to archive_num.txt
-kf to load archive_num.txt (Knotilus plaintext file)
-p to use Plink, [filename] to load filename
-s to use Seifert data (Note: data must be in quotes, i.e. as a string)
[-m] to use multiprocessing

Example:

>>> python corrterms.py 9x-1-38
9x-1-38
Downloading from Knotilus
Getting page source...... [DONE]
Getting page source...... [DONE]
took 8.10623e-06 sec to anneal
[[-3 -1  1  0]
 [-1 -4 -1 -1]
 [ 1 -1 -4  1]
 [ 0 -1  1 -3]]
H_1(Y) ~ Z/5ZxZ/15Z
Not using multiprocessing
Computed from quadratic form in 0.490264 seconds
7/10, 1/30, 13/30, -1/10, 13/30, 1/30, 7/10, 13/30, -23/30, -9/10, 1/30,
1/30, -9/10, -23/30, 13/30, 3/10, -23/30, -23/30, 3/10, 13/30, -11/30,
-1/10, -23/30, -11/30, -9/10, -11/30, -23/30, -1/10, -11/30, 13/30, -9/10,
-11/30, -23/30, -1/10, -11/30, 13/30, 3/10, -23/30, -23/30, 3/10, 13/30,
-11/30, -1/10, -23/30, -11/30, -9/10, -23/30, 13/30, 7/10, 1/30, 13/30,
-1/10, 13/30, 1/30, 7/10, 13/30, -23/30, -9/10, 1/30, 1/30, 3/10, 1/30, 5/6,
7/10, -11/30, -11/30, 7/10, 5/6, 1/30, 3/10, -11/30, 1/30, -1/2, 1/30, -11/30

This module handles double-branched covers and outputs the associated quadratic form.

Usage:

$ python graph_quad.py [-s] archive_num
$ python graph_quad.py -f archive_num.txt
$ python graph_quad.py -p [filename]

The first two usages are for Knotilus. Use -s to save the plaintext file in the current directory. Use -f to indicate loading the file 'archive_num.txt'. The printed time to 'anneal' has to do with how long the Knotilus server takes to render the link and create the plaintext file. The last usage is for PLink. Optional argument 'filename' indicates loading 'filename'. No optional argument starts the PLink editor.

The ndqf (negative definite quadratic form) module takes the negative definite quadratic form (as a matrix) and computes the correction terms. It does NOT check that the matrix you enter is negative definite. (The program itself currently checks this in the modules that create the matrix instead.)

Usage:

>>> x = NDQF([[-5, -2], [-2, -4]])
>>> x.correction_terms()
Computed from quadratic form in 0.04 seconds
'7/16, 1/4, -9/16, 0, -1/16, -3/4, -1/16, 0, -9/16, 1/4, 7/16, 0, -17/16,
-3/4, -17/16, 0'
>>> x.group
Structure decomposition: H_1(Y) ~ Z/16Z.
Generating vectors in order of invariant factor:
[[0 1]] has order 16.
Relation vectors (congruent to 0):
[[ 1 -6]]

The Smith module computes the Smith Normal Form of a numpy matrix, as well as the unimodular accompanying factors. The algorithm is an implementation of that of Havas and Majewski.

Usage:

>>> x = np.matrix([[-5, -2], [-2, -4]])
>>> d, (u, v) = smith_normal_form(x)
>>> assert u * d * v = x
>>> print d
[[1 0
  0 16]]