TeX Format Finite Biquandle Switch Lists

The lists below are presented in TeX display math mode so that they may be copied directly into a TeX document. 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 action 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 T, where <num-crossings> took values 2,...,6, S indicates that the source lists contain switches rather than sideways maps and -o T identifies that the output should be in TeX format.

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 -T, where -n 6 indicates that all lists up to order 6 should be created and -T identifies TeX format.

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