Package: khoca
Khoca is a packaging of Lukas Lewark's Khovanov homology calculatorkhocafor easy installation insage.From the original page:README for Khoca, a knot homology calculatorThe same text as in this README file is available (with illustrations) athttp://lewark.de/lukas/khoca.htmlKhoca is computer program to calculate sl(N)-homology of knots. The program has
been written for joint projects with Andrew Lobb such as [3, 4]. The paper [3]
also contains a description of the algorithm used by khoca. The main innovation
is to use Krasner's calculation of the sl(N)-homology of the basic two-crossing
tangle [2] for calculations of the homology of bipartite knots.Khoca calculates the following:Khovanov sl(2)-homology of arbitrary links, given as a braid or in PD code.Khovanov-Rozansky sl(N)-homology with N > 2 of bipartite knots, given by a
certain encoding of a matched diagram of the knot (see [3] and section
"Encoding of matched diagrams" below).Homology over the integers, the rationals or a prime field.Either equivariant homology, or homology with an arbitrary fixed potential.All pages of the spectral sequence of filtered homology over a field.Reduced and unreduced homology.Homology of sums and mirror images of knots.You are encouraged to contact me with any kind of questions or comments
regarding khoca. If you are using khoca for a project or publication, please
cite this web page, or the paper [3].InstallationBinaries for Linux are available for download fromhttp://lewark.de/lukas/khoca.htmlThey should run on any Linux installation that has python3.6. Binaries for
Windows or Mac are not available at the moment.The source code, including instructions on how to compile it, is available at
the GitHub repository khoca:https://github.com/LLewark/khocaTo run Khoca in Docker type:dockerrun-itsoehms/khoca:latestIts download size is 162 MB and it will need 516 MB of disk space on your
device. To create a new (resp. locally own) Docker image cd to the khoca
directory typedockerbuild-fDockerfile--tagkhoca:<your_tag>.If your machine has an older CPU it can happen that you getIllegal Instructionerrors. In that case you better should use the imagesoehms/khoca:old_cpu.UsageTo use the program, run khoca.py (a python3 script) from the command line.
khoca.py takes three arguments:The coefficient ring;0for integers,1for rationals, a primepfor the
corresponding finite field.A sequence of N integersa_0, ..., a_{N-1}separated by a non-digit
character, defining the Frobenius algebraF[X]/(X^N + a_{N-1}X^{N-1} + ... + a_0).  Alternatively,efollowed by a numberNfor equivariant computation
oversl(N). For example,-1.0.0gives the Frobenius algebraF[X]/(X3 - 1).A root of the polynomial given in 2. for the calculation of reduced homology
(for the dependence of reduced homology on a root, see [3]).  For example, to
get the standard graded reduced homology, use 0 as root. If you are not
interested in reduced homology, it does not matter what root you chose (and
khoca does not check that the number is actually a root).The option-pwill show progress bars,-vwill give more verbose
non-mathematical information, and-hwill print a short help text. Each
argument after the first three arguments, can be one of the following.BraidXcalculates homology of a link given as closure of the braidX,
formatted as in knotscape (a= first Artin generator,A= its inverse,b=
second Artin generator, etc.). This works only forsl(2)homology, otherwise
output is nonsensical.PdXcalculates homology of a link given in PD notation (as e.g. given on
KnotInfo). Again, this works only forsl(2)homology, otherwise output is
nonsensical.GaussXcalculates homology of a bipartite knot given as a matched diagram,
following the convention explained in the section below. This works forsl(N)homology for allN.MirrorXtakes the dual of the result at spotX.SumXYcomputes the homology of the connected sum of the results saved at
spotsXandY(numbers separated by a non-digit character).CalcXoutputs the result saved at spotX. If you forget this command, the
program will have no output.The program keeps a stack of computed homologies, enumerated 0,1,2... . Each of
the commands 1 - 5 puts a new homology on that stack, whereas the command 6.
prints the homology at a certain spot. This is mainly useful to compute
homology of sums of knots.Here are some examples:./khoca.py00.00braidaBaBcalc0calculates the classicalsl(2)Khovanov homology (both reduced and unreduced)
of the closure of the braidaBaB(knotscape notation), i.e. the figure-eight
knot../khoca.py0e20pd[[4,2,5,1],[8,6,1,5],[6,3,7,4],[2,7,3,8]]calc0calculates integral equivariantsl(2)homology of the figure-eight knot../khoca.py70.-10braidabcdefabcdefabcdefabcdefabcdefabcdefabcdefabcdef
calc0-pcalculates Khovanov homology of the(7,8)-torus knot overF7with perturbed
potential, displaying progress bars. This calculation takes roughly two
minutes, and shows that the spectral sequence does not collapse on the second
page, refuting the knight-move conjecture over finite fields (cf. [1])../converters/montesinos.py[1/5,1/3,-1/2]outputs[12,4,16,10,15,9,14,13], the code for a matched diagram of the(5,3,-2)-pretzel knot, aka the(3,5)-torus knot, aka10_{124}. So./khoca.py11.0.0.0.00gauss[12,4,16,10,15,9,14,13]calc0calculatesrationalsl(5)homology and the corresponding Rasmussen invariant of the(3,5)torus knot../khoca.py11.00braidaaadual0sum0+1braidaBaBsum2+3calc4calculatessl(2)homology of the sum of the trefoil, its mirror image and a figure-8-knot.Encoding of matched diagramsThis section describes how to encode a matched knot diagram, i.e. a diagram
that consists ofncopies of the basic 2-crossing tangle. Resolving each basic
tangle into two intervals and a chord results in a single circle with n
non-intersecting (red) chords, which may be on either side of the circle.
Enumerate the2nchord endpoints by walking around the circle. If a chord
connects the pointsiandj, letf(i) = j. Write down the listf(1), f(2), ..., f(2n)omittingf(i)iff(i) < i. Moreover, make the list entries signed, and
let the sign reflect the sign of the two crossings of the corresponding
2-crossing tangle. This list ofnnon-zero integers uniquely determines the
matched diagram. As an example, the standard diagram of6_1is matched and
encoded as[-4,6,5]."Half" of Montesinos knots are bipartite [3]. You may use the python3 script./converters/montesinos.pyto obtain the encoding of a matched diagram of
Montesinos knots.References[1] Bar-Natan: Fast Khovanov Homology Computations, Journal of Knot Theory and
its Ramifications 16 (2007), no.3, pp. 243-255, arXiv:math/0606318, MR2320156.[2] Daniel Krasner: A computation in Khovanov-Rozansky Homology, Fundamenta
Mathematicae 203 (2009), pp. 75-95, arXiv:0801.4018, MR2491784.[3] Lukas Lewark and Andrew Lobb: New Quantum Obstructions to Sliceness,
Proceedings of the London Mathematical Society 112 (2016), no. 1, pp. 81-114,
arXiv:1501.07138, MR3458146.[4] Lukas Lewark and Andrew Lobb: Upsilon-like concordance invariants from
sl(n) knot cohomology, arXiv:1707.00891.Lukas Lewark, 2018