Permutation TeX Format Finite Biquandle Switch Lists

The lists below are presented in TeX math mode format with the rows of each matrix written as permutations. Each list item is numbered as a birack, biquandle, rack or quandle the prefix BR, BQ, R or Q respectively together with the number of crossings as a superscript and an index as a subscript.

The up and down actions contained in the lists on this page correspond to the maps sa and sa. The lists are defined up to isomorphism: permutation symmetries (conjugation), inverse (interchange of classical crossing sign), interchange of the up and down action (orientation reversal), and combinations thereof, as described in reference [1] on the finite biquandles page.

For quandle related list items the corresponding related quandles are also shown. For example the biquandle BQ^5_{1} is quandle related, having it's up acton related to Q^5_{1} and down action related to the quandle Q^5_{1}. This is shown in the following lists as BQ^5_{1}(Q^5_{21},Q^5_{1}). Another example is birack BR^6_{7}, whose up action is not quandle related but whose down action is related to Q^6_{2}, this is shown as BR^6_{7}(-,Q^6_{2}).

Each list item is also identified as symmetric (S), as a pseudo-quandle (PQ), or a double pseudo-quandle (DPQ) wherever applicable. The two column invariants c_1 (the sum of the number of constant columns in U and D) and c_2 (the absolute difference of the number of constant columns in U and D are also shown.

Steps to calculate the above lists

The above lists were calculated from the plain format results using the following bash scripts

The script convert-lists converts the lists corresponding to a specified number of crossings into a specified format, the above lists were produced using commands convert-lists <num-crossings> -S -o PT, where <num-crossings> took values 2,...,6, S indicates that the source lists contain switches rather than sideways maps and -o PT identifies that the output should be in TeX format with rows written as permutations.

The script redo-dominant-pairs calculates the dominant pairs for crossings 2,...,6 and takes as a parameter the output format required. The above lists were produced using the command redo-dominant-pairs -n 6 -TP, where -n 6 indicates that all lists up to order 6 should be created and -TP identifies TeX format with rows written as permutations.

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