Biquandles of Small Size and some Invariants of Virtual and Welded Knots

4th May, 2015

This page contains links to the full set of lists, and associated software, described in [1] and amended in [2]. The lists are presented in three different formats, depending on your preference!

These are the lists of switch maps, S :

These are the corresponding sideways maps, F :

The lists are presented using the convention Xn={1,...,n} but the programmes that were used to create them are also able to handle the convention Xn={0,...,n-1} if preferred.

The switch list pages contain all the the essential pairs and essential singular pairs that have been discovered. These may be used by the braid programme to calculate invariants of welded knots and singular knots respectively. Using this approach a list of distinct non-trivial welded knots has been produced.

The lists were calculated using the following programmes:

The file biquandle-release-4-5-15.tar contains the C++ source for these programmes, together with a Makefile.

The programme nquandle is a search for finite quandles of size n that uses a "filling in blanks" algorithm based on the biquandle search algorithm of Sam Nelson and John Vo in the paper Matrices and Finite Biquandles (arXiv:math/0601145v4 April 2006). This approach creates a working list of prototype quandles, initialized to the zero matrix, and procedes sequentially to fill in zeros with members of the set Xn={1,...,n}, testing for contradictions with the quandle conditions. The quandle conditions also allow us to fill in additional blanks (zeros) in some cases but in others, a particular choice of an element from Xn leaves the prototype quandle incomplete. These incomplete prottype quandles are added to the end of the working list for further processing.

The command syntax for the programme nquandle is as follows:

Usage: nquandle -n=<n>[#[1|2|3]DrTXZ]
      n=<n> specify the size of the field
      #[1|2|3] generate debug output
        optionally specify level: 1=BASIC, 2=INTERMEDIATE, 3=DETAIL (default BASIC)
      D omit the down operation (=Identity) from output
      I do not remove inverses in symmetry check
      r conduct a rack search rather than a quandle search
      T display matrix output in TeX format
      X exclude summetry check
      Z number output matrices from zero

The programme nswitch is a search for finite biracks of size n that uses a similar filling in blanks algorithm as nquandle. The programme is able to initialize the working list with a set of U matrices read from a file. This allows us to seed the list of biracks calculated by nswitch with quandles so we only calculate biracks that are related to quandles.

The command syntax for the programme nswitch is as follows:

Usage: nswitch -n=<n>[#[1|2|3]RTZ <input_file>]
      n=<n> specify the size of the field
      #[1|2|3] generate debug output
        optionally specify level: 1=BASIC, 2=INTERMEDIATE, 3=DETAIL (default BASIC)
      R read seed patterns for U from <input_file>
      S use the S-Yang-Baxter relations rather than the F-Yang-Baxter relations
      T display matrix output in TeX format
      Z input and output X_n={0,...,n-1} rather than X_n={1,...,n}

The programme biquandle-search is a search for racks, biracks or biquandles that considers complete switches S=(U,D) where we consider either all permutations of Xn x Xn, or all combinations of U and D whose rows are permutations.

In addition, the programme biquandle-search is able to read in a list of biracks from an input file and test whether they are racks or biquandles. It is able to remove isomorphism symmetries from a list of biracks, compare biracks from one input file from another, and is able to remove biracks in a file either from a calculated list or from a list read from another file. The programme is capable of searching for switches or sideways maps, defaulting to searching for sideways maps. The S options controls this element of the programme. It is the biquandle-search programme that has been used to calculate most of the lists presented above.

The command syntax for the programme biquandle-search is as follows:

Usage: biquandle-search -n=<n>[#[1|2|3]BbCcDefIiPQqRrSsTXxZ [<input_file_1>] [<input_file_2>]]
      n=<n> specify the size of the field (must be provided)
      #[1|2|3] generate debug output
        optionally specify level: 1=BASIC, 2=INTERMEDIATE, 3=DETAIL (default BASIC)
      B display matrix output in braid programme switch format
      b birack (not biquandle) search
      C convert the specified Z option on input the opposite on output
      c compare search results with <input_file_2> (note <input_file_1> need not be present)
      d doodle switch search (may be combined with b to avoid testing the biquandle condition)
      D display related quandle information as part of TeX output (requires quandle list in <input_file_2>)
      e exclude anything isomorphic to an item in <input_file_2> (turns on c, <input_file_1> need not be present)
      f produce sideways maps from the S-biracks in <input_file_1> (turns on R)
      I only run code investigating Y-B equation
      i identify indecomposable racks only
      P display rows as permutations
      p check rows of the list read from are permutations
      Q search for quandle relatives in calculated list (may be used with R)
      q search for quandles (turns on b and R)
      R read data from <input_file_1>
      r search for racks (turns on b and R)
      S S-Yang-Baxter search
      s start search from birack conditions not switch conditions
      T display matrix output in TeX format
      X exclude summetry check
      x exclude permutations from the isomorphic symmetry check
      Y only run code investigating Y-B equation
      Z input and output X_n={0,...,n-1} rather than X_n={1,...,n}

The programme dominant-pairs takes as input a file containing a list of biquandles and produces lists of dominant pairs, separated into virtual pairs, weld pairs and essential pairs. It is capable of producing either a single output file or separate files for each type of dominant pair. Various output formats are supported, including formatting dominant pairs suitable as input switches for the braid programme. The essential pairs yield welded knot invariants and were used to produce a list of distinct non-trivial welded knots.

The command syntax for the programme dominant-pairs is as follows:

Usage: dominant-pairs -n=<n>[#[1|2|3]BPsTZ] <input_file>
      n=<n> specify the size of the field
      #[1|2|3] generate debug output
        optionally specify level: 1=BASIC, 2=INTERMEDIATE, 3=DETAIL (default BASIC)
      d doodle search, requires S-candidates in and T-candidates in
      B display output formatted as input switches for the braid programme
      P display rows as permutations
      s creates a single output file rather than separate ones
      T display matrix output in TeX format
      U swap the weld conditions, so weld crossings are welded to the underside of the plane
      Z input and output X_n={0,...,n-1} rather than X_n={1,...,n}
      <input_file> specify the file containing the source biquandles

The input files used to calculate the lists presented on the results pages were as follows. The steps used to determine the content of these files are described on the plain format switch results page.

The following files contain the lists of essential welded pairs and essential virtual pairs, written by dominant-pairs in a format suitable for use as input switches for the braid programme:

References:

[1] A. Bartholomew and R. Fenn. Biquandles of Small Size and some Invariants of Virtual and Welded Knots (arXiv:1001.5127v1).

[2] A. Bartholomew and R. Fenn. Biquandles of Small Size and some Invariants of Virtual and Welded Knots Corrigendum (to appear).

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