lattice - a computational tool for lattice cohomology
lattice [file1 [file2 ..] ]
lattice can compute the root of the zeroth lattice homology of a singularity given the intersection form of the link, a list of bad vertices and the values of the canonical divisor on those vertices. The arguments should be names of files which the program parses. The files should contain a list of commands which the program executes. Next the user is provided with a prompt to input more commands. Commands are given by a single letter in the beginning of lines. The rest of the line may contain parameters, or it is ignored if it contains more than the expected parameters.
i
Input data. The character following i should be one of the
following:
|
n |
followed by an integer to enter n, the number of vertices. | ||
|
v |
followed by an integer to enter nu, the number of bad vertices. | ||
|
I |
followed by n^2 integers to enter I, the intersection matrix. This should be preceded by entering n. | ||
|
b |
followed by nu integers to specify which vertices are bad. This should be preceded by entering nu via v. | ||
|
K |
followed by nu integers to specify the values of the anti-canonical divisor Z_K on the bad vertices. This should be preceded by entering nu via v. | ||
|
f |
followd by an integer to specify output format. If a nonzero value is entered, then the program writes the values of chi into the file ’output’. This file is owerwritten each time, so make sure to move it somewhere else if you want to keep it. |
p
Print data. The p should be followed by one of the
characters nvIbK to print data obtained via the input
command or r to print the root.
c
Calculate root from data provided.
f
Flush data, that is, forget the root if it has been
calculated.
q
Quit the program.
The program calculates the minimal path cohomolgy only in the reduced lattice. This means that the minimum ranges over only a particular set of paths. In practice, we have not found this minimum to be bigger than the actual minimum (assuming we have calculated the actual minimum) but this reduced minimum has not been proved to coincide with the actual one in general.
Since any path gives an upper bound for the geometric genus, it follows that if the reduced minimal path cohomology produced by the program coincides with the geometric genus of some analytic structure with the given graph, then this is really the minimal path cohomology. This is the case e.g. for Newton nondegenerate hypersurface singularities.
The files data1, data2 and data3 are examples which calculate the roots of one topological type each.
The lattice cohomology was introduced by Némethi András as an invariant of links of surface singularities. The main theoretical ingredient here is Tamsás László’s reduction theorem and a generalized form of Laufer’s computation sequence. The program was made by Baldur Sigurðsson, baldur at renyi dot hu, started in 2013.