MathematicalSoftwareAndFreeDocuments/06

数学ソフトウェアとフリードキュメント/06

MathematicalSoftwareAndFreeDocuments VI †

• 2008 March 22th (Sat 13:00 -- 18:00)
• Main campus of Kinki University, 38th building, 2F, Conference Room No.9 (Campus map)
• Organizer: Noro, Masayuki (Kobe University) noro at math.kobe-u.ac.jp, Hamada, Tatsuyoshi (Fukuoka University) hamada at holst.sm.fukuoka-u.ac.jp

Speakers †

• Deguchi, Hiroaki
• Kanamitsu, Akiko
• Kiriu, Yusuke
• Ochiai, Mitsuyuki and Kako, Fujio
• Ohtsuka, Koji
• Ohshima, Toshio

Program †

Abstract †

Mathematica Player is an innovative new take on viewer applications, and Mathematica Player Pro is the low-cost professional platform for running interactive Mathematica applications and documents.

In my talk, I will introduce Mathematica6 new feature, The Wolfram Demonstrations Project which has over 1,000 examples of its interactive capabilities, and new Mathematica Player familiy.

There are many Finite Element Solver of partial differential equations by finite element method. However they use mathematical knowledges as black box. If we want teach or study variational method to solve partial differential equations, software (with hidden mathematical knowledge) cannot help our strength of mathematics. So the specialty of FreeFem++ has the mathematical expression and the mathematical knowledge help FreeFem++ programming. In my talk, we show how to use and what problems can be solved by FreeFem++. The site of FreeFem++ is: http://www.freefem.org/ff++/

MathBlackBoard is a prototype of intuitive formula editor which operation is based on "drag and drop". I will introduce basic operations of the software.

My specialty is Art. I talk about Mathematical software, Frontend technology and Art.

1. Implementation of some function about knot theory. (with Takuya Sakasai, Tokyo Univ.)
2. Reference document of Knot Data mounted on Mathematica 6.0 (with Koji Ouchi, Wolfram Reserch Inc.)
3. Project of mathematical Universal Frontend Developers.
4. Symbol manipulation environment in the art.
5. Research of clothes using symbolic manipulation. (with Takuya Kitamoto,Yamaguchi Univ.)

This program SKB consists of the following three parts:

1. SnapPea programmed by Jeff Weeks which can do to compute manifold invariants including hyperbolic volumes, isometries, and fundamental groups and etc..
2. K2K [1] programmed by M. Ochiai and Noriko Imafuji which can do to compute many polynomial invariants of knot and link using skein relations or matrix representations by Hecke algebras.
3. btd [2] programmed by M. Ochiai Noriko Morimura which can do base tangle decompositions of two n-tangles with 1<n<10.

This also icludes three input tools and one output tool.

1. The main tool "KnotInput" is a one to input knots and links by a mouse tracking method.
2. The second tool "BraidInput" is a one to input closed braids as alphabets such as "aCbCbbCA^4bb" by keyboard and later users can append another knot components by mouse tracking method.
3. The third tool is to input knots and links including in text files as p-data format (****.prd) or snappea deta format (****.spp).
4. Users can show knots and links as OpenGL drawing figures based on p-data format by "KnotViwer".

This program uses Java as user interface library and most of them are written by C lanuage with OpenGL-GLUT graphics library. And also it works on MacOSX on Intel Macintosh (or Linux on Intel machines) and Windows XP and Vista operating system.

Main window bases mainly on file access. In this case, users can compute sequentially invariants of many knots and links in a file. It consists of many menu including of "FILE","Input a knot and link", "SnapPea", "K2K on p-data", "K2K on braid data", "btd", and "Help & About". "FILE" has sub-menus "OPEN", SAVE", SAVE AS", CLEAR", "EDIT a text", and "EXIT". "Input a knot and link" has sub-menus "Input by mouse tracking", "Input from a file of type K2K", "Input from a file to generate all projections", "Input by File of type SnapPea", and "Filtering about triviality and mutation". "SnapPea" has sub-menus "fundamental groups of complements", "fundamental groups of manifolds", "volumes of complements", "volumes of manifolds", "representations of complements about Zn", "representations of complements about Sn", "isometry", "symmetry groups of complement", "amphiceiral", "invertible", and "group presentation". "K2K on pdata" has sub-menus "HOMFLY polynomials", "Kauffman (& Q) polynomials", and "Pdata to braids". "K2k on braid data" has sub-menus "Alexander polynomials of braids", "Jones polynomials of braids", "Parallel HOMFLY polynomials of braids", and "Inout a braid". "btd" has a sub-menu "A base tangle decomposition".

On the other hand, users can do invariants of only one knot and link by "KnotInput" or "BraidInput". These two programs can work also as stand-alone programs (but the second program does not work on practice, because it must accept a string as a braid from a Java interface window) and have three output files "lastPldata.pl", "LastSnappea.spp", and "pdata.prd". "lastPldata.pl" and "LastSnappea.spp" are re-drawing data and "pdata.prd" includes combinatorial text data of knots and links. In particular, users can use "lastPldata.pl" (resp. "LastSnappea.spp") as an input data in K2K on Mathematica (resp. another SnapPea program such as on Linux or Windows XP). In these programs, users can save the current data by keying "w" or "W" as the file "lastPldata.pl" and reload by keying "r" or "R" from "lastPldata.pl". When quiting the programs, the current knot data will be saved to the file "pdata.prd" which is the default text data file to compute. "KnotInput" and "BraidInput" have the following same pop-up menu: "SnapPea Invariants of a complement" has sub-menus "Solution Type", "Volume", "Fundamental group", "Fundamental group of (1,-1) surgery", "Two bridge structure", "Group Representation to Z_3", "Group Representation to Z_5", "Group Representation to S_3", "Group Representation to S_5", "Normal Surface structure", "Symmetry Group", "Isometry: The first", "Isometry: The second", and "Replace the second to the first (C)". "(p,q) Dehn Surgery " has no sub-menu but later when inputting surgery coefficients, users will get the same sub-menu like as "SnapPea Invariants of a complement". "Compute a polynomial" has sub-menus "Jones polynomial", "HOMFLY polynomial", "Alexander polynomial", "Conway polynomial", "Kauffman polynomial", and "Q-polynomial". And there exit another menus "Replace the second to the first (S)", "MirrorImage". "Read a p-data", "Write a p-data", "Clear", and "Quit".

Users can start this software to double-clicking "SNB.jar" and access directly files in the folders /PLDIR(pl-data), /PDATADIR(p-dat), Isometry(triangulation data of knots and links), /SIsometry(triangulation data of surgery manifolds), /samepolynomial(files including pdata with same HOMFLY polynomials), and /btd(base tangle decompositions), but recommend not to touch files in the folders /Braid_9, /Braid_12, and /Braid_15(Hecke algebras matrix representations).

Users can divide any files of pdata format into another files each of which contain pdata with the same HOMFLY polynomials in the folder /samepolynomial by a series operation consisting of "Filtering about triviality and mutation"-> "action" -> "filtering by polynomial invariants". Later users can use these files as an input file to filter about mutation by by the item "filtering about triviality or mutation". Also users can recognize knots, in which are in any files of pdata format, to be trivial by the item "filtering about triviality or mutation".

In this software, the default data format is a pdata format. Given an oriented knot projection diagram D of a knot K, at first a point s on K is fixed in D, and you give each crossing point a sign + or - at the standard manner, and then from the point s along the orientation you assign a number sequentially from 1 to 2n to each crossing point, where n is the total number of crossings. Then starting at s, you write down a signed number of j-th under crossing point when reaching at i-th over crossing point.For example, the following gives a pdata of a knot K, where K has 9 crossing points:

 18 1
 18
 -9 -13 -17 -1 -15 -3 -11 -5 -7

Users can execute base tangle decompositions of any knots and links using the menu "btd":You click sequentially "btd"->"A base tangle decomposition" and you will get the main window of btd. At first, you determine the string number r of two r-tangles (default is 2) to press the key "2","3",..."9". Note that you can do also a mutational operation to press the key "m", "h", and "v" (standard mutation, horizontal muation, and vertical mutation). When you input the first r-tangle, you must start node 0 at upper-left side on the circle. In the case, a small rectangle indicates the direction. By clicking the menu "The Base Tangle Decomposition of n-tangle", you can do it. Then you can input the second r-tangle freely but the orientation must compatible with the first. Also By clicking the menu, you can do it. At final, you can compute the HOMFLY polynomial of the resulting knot or link by clicking the menu "HOMFLY polynomial of two n-tangle (1<n<10)"->"Compute" or "Compute or Draw". When you press the latter, you will get a figure on the main window of "KnotViewer", then you can compute many invariants like as "KnotInput".

It will noticed that the tool given by the menu "Two bridge structure" does not work correctly in the current version for may be SnapPea's bug?.

[1] Computer Aided Knot Theory using Mathematica and MathLink, N. Imafuji and M. Ochiai, Journal of Knot Theory and Its Ramifications, Vol. 11 No. 6, 945-954, 2002.

[2] Base tangle decompositions of n-string tangles with 1<n<10, M. Ochiai and N. Morimura, to appear in Experimental Mathematics, 2008.

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