knot program

Home page of K.Kodama.

Current version: executable set for LINUX
knotGTK, port to Windows by Toshio SUMI
Use LINUX emulation on MacOSX.

KNOT program
- overview
- Features
Overview

The software "KNOT" is a tool for knot theory to draw diagrams of knots and links(in S^3) and knotted surfaces(in S^4) and calculate many invariants.

If you have any question or problem, please do not hesitate to contact me(kodama@math.kobe-u.ac.jp). Your reporting will be good contributions to improve the computer program. Thank you,
- Down load: ftp://ftp.math.kobe-u.ac.jp/pub/knot/.
- Executable binary: Knot.bin.(DATE).tar.gz
- Source codes: Knot.src.(DATE).tar.gz
  GNU/sather-1.3.2 compiler is also needed.
- OS/Environments:
  - Original version is verified on LINUX(e.g. RedHat,Debian,SuSe,Vine,Plamo) with X Window System.
  - Port to Windows, MacOS X and Solaris are available by Toshio SUMI.
    Please report to him if you think partings be helpful.
- License: Free and Open under the term of GNU General Public License(GPL).
Features

New
1. Meridian generators are displaied on diagram when [Invariant]-[Group] selected.(2007-04)
2. Bug fix: A function for Artin normal form fails to make normal form.(2006-02-15)
3. Keyboard shortcut(2005-12-01, experimental)
  See Doc/knot.Xdefs for key binding.
4. -logfile option(2005-11-29)
5. -log option (2005-11-28)
6. Bug fix: Some times it fails to compute Alexander ideal.(2005-11-28)
7. [Effects on Diagram]-[Braid] accepts new option switch. It changes closed braid or (open) braid.(2005-05-16)
8. -cov and -sl2 are splited to sub options. (2004-11-13)
9. "pstricks" graphical commands is used for LaTeX picture by default. (2004-11-13)
  In old version, "tpic" special commands is used.
Suggestion
Almost results of Invariants options are LaTeX format. If you want putting it to other programs(e.g. Mathematica), make appropriate filter of gawk/perl/ruby/python to convert styles.
Example:
script:
```
cat input_file | gawk '{gsub("{","(");gsub("}",")");print $0}'
```
input_file:
```
z^{-2}*(-v^{6}+4*v^{4}-5*v^{2}+2)
+(-v^{6}+8*v^{4}-13*v^{2}+6)
+z^{2}*(v^{6}+4*v^{4}-14*v^{2}+5)
+z^{4}*(-7*v^{2}+1)
+z^{6}*(-v^{2})
```
Result:
```
z^(-2)*(-v^(6)+4*v^(4)-5*v^(2)+2)
+(-v^(6)+8*v^(4)-13*v^(2)+6)
+z^(2)*(v^(6)+4*v^(4)-14*v^(2)+5)
+z^(4)*(-7*v^(2)+1)
+z^(6)*(-v^(2))
```
Invariants
1. knot group G(K) of Wirtinger presentation for knot,link in S^3, knotted surface in S^4
2. knot group for theta-3 curve
3. Signature
4. Goerits matrix and knot value
5. Alexander polynomial(1-variable) for knot,link in S^3
6. test cyclic period using Alexander polynomial for knots
7. Alexander/Elementary ideal(1-variable) for knot,link in S^3, knotted surface in S^4.
  1. Alexander matrix from free differential
  2. Alexander matrix (simplified).
  3. Elementary ideals.
  4. Novikov(extended Alexander) polynomial: For i-th ideal, Novikov (extended Alexander) polynomial A(i) is the generator of the ideal as a positive infinite power series. Note that A(i)/A(i+1) called Novikov index.
8. Alexander polynomial(multi variable) for link in S^3.
9. Alexander/Elementary ideal(multi variable) for knot,link in S^3, knotted surface in S^4.
  1. Alexander matrix from free differential
  2. Alexander matrix (simplified).
  3. Elementary ideals.
10. 3-variable Alexander polynomial for theta-3 curve.
11. Conway polynomial.
12. Jones V polynomial.
13. Jones V polynomial(parallel version).
14. HOMFLY P polynomial.
  1. Change forms "P(v,z)", "P(l,m)","P(x,y,z)"
15. HOMFLY P polynomial(z-lowest part).
  1. Change forms "P(v,z)", "P(l,m)","P(x,y,z)"
16. Kauffman F polynomial.
17. Q polynomial.
18. representation G(K) to S(n) (permutation of n-set) for knot,link in S^3, knotted surface in S^4.
  1. search all n-fold coverings(with equivalent coverings excluded)
  2. meridian/longitude system
  3. branch and peripheral data
  4. covering linkage
  5. Homology group H1 for branch/un-branch covering
  6. Homotopy(Fundamental) group pi_1 for branch/un-branch covering
  7. twisted Alexander polynomial.(1-variable)
  8. twisted Elementary ideals.(1-variable)
  9. twisted Novikov(extended Alexander) polynomial.
  10. covering distribution (statistics) invariants
    1. statistics of Yang (orbit) diagram.
    2. statistics of Branch index.
    3. statistics of Covering linkage.
    4. statistics of H1(branched cover).
    5. statistics of H1(unbranched cover).
19. representation G(K) to sl2(p) for knot,link in S^3, knotted surface in S^4. (p:prime integer)
  1. search all representations (with equivalent representations excluded)
  2. twisted Alexander polynomial.(1-variable)
  3. twisted Elementary ideals.(1-variable)
Graphical (modal) edit options
1. "New knot": clear screen and input new diagram
2. "Add Compo": Add components/string
3. "Crossing": crossing change
4. "Move vertex": Move vertex of strings
5. "Cut": Cut off segment in diagram
6. "Del string": remove string
7. "Add Band":
  1. saddle band for knotted surface in S^4.
    In this version, it cannot distinguish upper or lower saddle, So, we can not draw exact geometric condition but compute knot group and related invariants.
  2. band for surgery
  3. 3rd element for theta 3 curve
8. "Band surgery": surgery along band
9. "Invert string": invert string/component
10. "Shift Diagram": move diagram in the screen
11. "Jump(over)", "Jump(under)": over/under jump move
Graphical effects
1. "Braid": show/input braid word
2. "n-Data": show/input n-Data word
3. "Torus link": input parameter of torus link T(p,q)
4. "2-Bridge": input parameter of 2-bridge link Schubert's notation S(p,q)
5. "2-Bridge": input parameter of 2-bridge link Conway's notation C(a1,a2,...,an)
6. "Pretzel": input parameter of pretzel link P(a1,a2,...,an)
7. "Alternate": make crossings alternating
8. "coherent band": make components coherent along bands
9. "magnify x2": magnify the diagram 2 times
10. "magnify x1.4":magnify the diagram 1.4 times
11. "magnify x0.7":magnify the diagram 0.7 times
12. "magnify x0.5":magnify the diagram 0.5 times
13. "mirror-X": mirror along YZ plane, so invert X-axis
14. "mirror-Y": mirror along XZ plane, so invert Y-axis
15. "mirror-Z": mirror along XY plane, so invert Z-axis. (crossing change)
16. "rotateR": rotate by 90 degree
17. "-rotateR": rotate by -90 degree
18. "Back": backward edit history
19. "Forward": forward edit history
20. "force put history": put current diagram into history
Input, Output and Logging options
1. "Load knot": load a knot-data
2. "Save Knot": save current diagram as a knot-data
3. "Save Braid": save as a braid word
4. "Save n-data": save as a n-Data
5. "Save Knot ver.1": save as knot-data(ver.1). (old format)
6. "log file: ": on/off logging results
7. "log LaTeX picture": logging diagram as LaTeX picture
8. "log Knot": logging diagram as knot-data
9. "log Braid": logging diagram as braid word
10. "log n-Data": logging diagram as n-Data
Other options
1. "Smooth Draw: ": switch for display diagram as smooth/rigid picture
2. "Lattice move: ": switch for input diagram on lattice points or not
3. "Show restrictions": show restriction of current version
4. "Show knot data": show knot-data of current diagram
5. "Check knot data": check knot-data of current diagram
6. "Show tcode": show inner representation of crossing data
7. "Show tcode theta": show inner representation of crossing data for theta curve
8. "Show braid": show braid word of current diagram
9. "Show ndata": show n-Data of current diagram
10. "experimental feature": experimental feature for development
Keyboard shortcut
Many of GUI menu operations are usable from keyboard shortcut. Key bindings are defined as 2-stroke key actions in Doc/knot.Xdefs.
These definitions are experimental. And it may be changed. Check your definition file Doc/knot.Xdefs.
Empty action:
When you confused key stroke, hit space to break off.
```
" ": empty_proc
```
Effects:
```
"e2": Bridge2
"ea": Alternate
"eb": Braid
"ec": Coherent
"el": Mag2
"em": Mag14
"en": nData
"eo": Rot
"ep": Pretzel
"er": Mag14R
"es": Mag2R
"et": Torus
"ex": MirrorX
"ey": MirrorY
"ez": MirrorZ
```
File in/out:
```
"fb": SaveBraid
"fl": LoadKnot
"fn": SaveNData
"fs": SaveKnot
```
History:
```
"hb": BackHist
"hf": FwdHist
"hs": SaveHist
```
Invariant:
See section of "p*" for polynomial invariants.
```
"ia": AlexIdeal
"ib": AlexIdealM
"ic": Covering
"ig": GoeritzMat
"ik": KnotGroup
"il": sl2p
"ip": CyclicPeriod
"is": Signature
```
Logging:
```
"lb": LogBraid
"lk": LogKnot
"ln": LogNData
"lp": LaTeXPicture
"ls": LogSw
```
Edit Mode:
```
"ma": AddCompo
"mb": AddBand
"mc": Crossing
"md": DelString
"mf": ShiftDiagram
"mi": InvertStr
"mm": MoveVert
"mn": NewKnot
"mo": JumpOver
"ms": BandSurgery
"mt": Cut
"mu": JumpUnder
```
Other:
```
"ol": LatticeMove
"os": SmoothDraw
```
Polynomial:
```
"pa": AlexPoly
"pb": ApexPolyM
"pc": ConwayPoly
"pf": Fpoly
"pp": Ppoly
"pq": Qpoly
"pv": Vpoly
"pw": VpolyPara
```
Theta curve:
```
"ta": ThetaAlexPoly
"tg": ThetaGroup
```
Quit:
```
"zz": QuitProc
```
GUI options
We can give start up options as GUI tool.
1. -gui : to force GUI, put this at the top of options as "knot -gui -log"
2. -log : enable logging at boot time.
3. -logfile (file) : set file name to log out.
  knotLog.(date) by default.
Scripts and start up options as a command tool
1. Use helper script "knot.sh" if you need.
2. For automated computation about a list file of knots, use additional script "knot_dir.sh".
3. -help: print out help message and exit
  e.g. knot -help
4. as a stream filter: -s (options)
  read knot-data from standard in-stream and output to standard out-stream
  e.g. knot -s -ai -p < knot_data
  Options are executed in sequence from left to right. The option -s must be the first one.
5. or explicitly read a file: -if (input_file) (options)
  read knot-data from "input_file" and output to standard out stream
  e.g. knot -if k3 -ai -p
  Options are executed in sequence from left to right. The option -if must be the first one.
6. options for invariants:
  1. -a: Alexander polynomial
  2. -ai: Alexander/Elementary ideal
  3. -am: multi variable Alexander polynomial
  4. -ami: multi variable Alexander/Elementary ideal
  5. -c: Conway polynomial
  6. -v [n]: Jones polynomial(n-parallelized)
  7. -p [type]: P(HOMFLY) polynomial. type=vz,lm,xyz. (vz by default)
  8. -p_low [type]: The lowest term of P as a z variable polynomial. type=vz,lm,xyz. (vz by default)
  9. -f: F polynomial
  10. -q: Q polynomial
  11. -cyc: test cyclic period using Alexander polynomial
  12. -cov [n] [sub options]: n-fold covering(n=2 by default)
    Sub options:
    1. br : branch indices.
    2. lk : covering linkage.
    3. h1b : H_1 as branched covering space.
    4. h1u : H_1 as unbranched covering space.
    5. gp : Group relations.
    6. ai : Twisted Alexander invariants.
    7. distr: Distributions.
  13. -sl2 [n] [sub option]: representation to SL2(p) (p:prime,2 by default)
    Sub option:
    1. ai : Twisted Alexander invariants.
  14. -grp: knot group
  15. -kv: Goeritz matrix and knot value
  16. -sign: Signature
7. options for conversions:
  1. -alt : alternate
  2. -mirror [axis] : mirror. axis=x,y,z. (x by default)
  3. -rot: rotate by 90
  4. -rotn: rotate by -90
  5. -mag [m] : m-magnify(m=2.0 by default)
8. options for output data
  1. -knot: knot data
  2. -braid [sub option]: braid word Sub options:
    1. a : Artin normal form
    2. ra : reduced Artin normal form
  3. -ndata: n-data
  4. -picture [width [height]] [sub options]: LaTeX picture. Measure by millimeter
    Sub options:
    1. s : smooth curve
    2. r : rigid line (default)
    3. tpic : LaTeX tpic
    4. ps : LaTeX pstricks (using psbezier when "s" is enabled) (default)
```
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