Alexander Stoimenow

Some programs

Here is C++ source code of some programs. My programs require the GNU C++ compiler g++ (version 2.7.2 f.) and a UNIX shell. Since even the GNU compiler does not always handle correctly templates and offers internal compiler errors, I cant exclude funny results. Please report any errors and problems you have!

A program for solid torus invariants and PostScript images

Here is a gzipped tar-file

gsinv.tgz

of some programs which can be used to calculate the Fiedler solid torus invariants, generate PostScript images of knots, links, and tangles (1/10 in size of those rendered by KnotScape and 1/100 of those rendered by the Adobe Illustrator), and convert between different formats (including the one of KnotScape's editor).

More information on what the programs do and instructions how they are to be used may be found in the paper

Th. Fiedler and A. Stoimenow, New knot and link invariants, Proceedings of the International Conference on Knot Theory "Knots in Hellas, 98", Series on Knots and Everything 24, World Scientific, 2000.

The following papers and monographs use essentially calculations with these programs:

Th. Fiedler, Gauss Diagram Invariants for Knots and Links, Kluwer Academic Publishers, Mathematics and Its Applications Vol 532 (2001).
Th. Fiedler, Gauss diagram invariants for knots which are not closed braids, Math. Proc. Cambridge Philos. Soc. 135(2) (2003), 335--348.
A. Stoimenow, Mutant links distinguished by degree 3 Gauss sums, Proceedings of the International Conference on Knot Theory "Knots in Hellas, 98", Series on Knots and Everything 24, World Scientific, 2000.

Remarks:

Compiling the main program may, however, take several minutes.
To run the program lsd2sp requires also GNU's AWK program gawk installed on your system.

Odd crossing amphicheiral knots

A tar-file with 5 programs (and several input/data files) written for calculations in the proof of the odd crossing amphicheiral knots.

oddach.tar

The test messages what is computed compare well (not perfectly) to this version of the paper.

Program aqv23tst7.C is for $15+4k$ crossings (though it was updated to subsume the step of Jones polynomial calculation explained in the paper). Programs aqv23tst8*.C are for $17+4k$ crossings.

Compiling with g++ on UNIX and calling the program without any argument should work (though some programs give no output, because there are no diagrams of the sought type). There are some comments in the programs, but better let me know if you like to try them out. So far I give no detailed explanation, since I'm not sure who will be interested...

Hoste's conjecture for 2-bridge knots

h2br_m2.nb, MATHEMATICA notebook with complete calculation (compiled with version 10.2). Contains comments and some extraneous calculations. (If you follow the comments you will be able to redo all needed calculations, although a bit of fiddling may be necessary.)

alpha_bd.m, and almax_bd_7a.m, two data files which can be read in from the notebook file to avoid the most time consuming parts of the calculation.

h2br_m2.pdf and h2br_m2.ps.gz, printouts of the MATHEMATICA notebook (with full graphics). PS is compressed because it's 63M.

h2br3a.ps and h2br3a.pdf, draft of the paper, not in publication-ready form, but with full explanation of all details of the calculation. Comments in the MATHEMATICA notebook refer to this draft.

Programs for knot markings (maximal generators), the sl_N polynomial, and hyperbolic unzipping of graphs

This tar file

markgs.tar

contains a program illustrating the calculation of the number of non-isomorphic knot markings of graphs (per convention in this section always 3-connected planar cubic, unless stated otherwise), which gives the number of maximal knot generators by crossing number. The archive is supplemented with data for genus 5. It contains the following files.

markgs.C: the main C++ program. Its call is
```
markgs <gr_file> <aut_file>
```
where gr_file: is a file of 3-valent 3-connected planar graphs and aut_file contains data of their automorphisms (format see below).
*.h: header files that are needed to compile the various C++ programs
tri.18: the list of 3-connected planar cubic graphs with 18 vertices as output by plantri in ASCII format (-a option).
Note that for generators of genus g one needs n=4*g-2 vertices.
asc2math.awk: a small GNU AWK script that converts plantri's output into a sequence of integers.
tri.18.aut2: the list of automorphisms (symmetries) of the graphs in tri.18 as computed by MATHEMATICA.
tri.18.stat: the output of
```
markgs tri.18.g tri.18.aut2
```
tri.18.stat2: the sum of the vectors in tri.18.stat, giving the numbers of maximal knot generators of genus 5 by crossing number, as seen in Table 4 of my paper "On dual triangulations of surfaces, Lie algebra invariants, and alternating link volumes" here

Use

gawk -f asc2math.awk tri.18 | sed 's/[^0-9][^0-9]*/ /g' > tri.18.g

to move the plantri format of graphs to one where each graph of n vertices is encoded by a sequence of 3*n integers, with each triplet giving the vertices adjacent to a given vertex 1,...,n in cyclic order of the (unique!) planar embedding. The sequence of 3*n integers is preceded by n (the number of vertices) and an integer id.

The file tri.18.aut2 contains the automorphisms of the graphs in tri.18, with each line standing for one graph. The line starts with a non-negative integer s, which means that the automorphism group has order s+1. (The trivial automorphism is always discarded.) Then there is a sequence of s*n integers, with each subsequence of length n being the permutation of the n vertices of each non-trivial automorphism. These automorphisms were generated with MATHEMATICA.

The first entry in the vectors of tri.18.stat corresponds to maximal generators of crossing number 6*g-3(=27), and such do not occur here (albeit they do occur for g≥6). The last entry in the vectors corresponds to crossing number 10*g(-6)=44, and is also always 0 (for every g). There is a single graph where the vector is 0, i.e., there are no knot markings at all. (This graph is called B₃ in the paper.)

The archive further features:

gr2dt.C: a program that unzips a graph G along a perfect matching and outputs the Millett-Ewing notation of the resulting link (called L'_G in the paper). Its exterior is the gluing of two ideal rectangular (hyperbolic) polyhedra whose 1-net is the median graph of G. Call by
```
gr2dt < tri.18.g > tri.18.ken
```

markgs_X4f5c.C: the fastest implementation I have (as of 5/4/25, using extensive cut optimization) for computing the sl_N polynomial. Call as

plantri -a -d 19 -c4 | head | perl -f asc2math2x.prl2 | cat -n | markgs_X4f5c

and you should get

34 1  3 19 59024 0 -42732 0 -11966 0 -25044 0 16510 0 3734 0 436 0 36 0 2 
34 2  3 19 62592 0 -50976 0 5992 0 -35202 0 13578 0 3554 0 424 0 36 0 2 
34 3  3 19 -12128 0 7696 0 14930 0 -30664 0 15180 0 4430 0 512 0 42 0 2 
34 4  3 19 15904 0 -8720 0 4466 0 -31588 0 15252 0 4142 0 500 0 42 0 2 
34 5  3 19 17888 0 -59312 0 77854 0 -49428 0 9044 0 3474 0 440 0 38 0 2 
34 6  3 19 21952 0 -29072 0 38920 0 -46128 0 10372 0 3494 0 422 0 38 0 2 
34 7  3 19 44384 0 -68648 0 20508 0 -18938 0 18048 0 4136 0 470 0 38 0 2 
34 8  3 19 40736 0 29088 0 -51298 0 -40782 0 17212 0 4486 0 514 0 42 0 2 
34 9  3 19 -89824 0 6672 0 85010 0 -22074 0 15484 0 4186 0 502 0 42 0 2 
34 10  3 19 15536 0 8964 0 6204 0 -44626 0 10312 0 3184 0 388 0 36 0 2 
ended at 34 10

which are the sl_N polynomials of plantri's first 10 cubic planar c4c graphs with χ=-17.

NOTE: the input format is as above, cyclic ordering of adjacent vertices, but this program works also for non-planar graphs!

asc2math2x.prl2: a variant of asc2math.awk for PERL since plantri outputs for large vertex count non-printable characters (ASCII code above 126) and GAWK seems to suffocate with them

Alexander Stoimenow,
stoimeno_stoimenov.net

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