Virtual pairs

There are many (isoclasses of) virtual pairs, for example, if q is a biquandle and a\in Aut(q), then the asignement

(x,y)\mapsto (a^{-1}(y),a(x))

is an involutive biquandle, that we call i_a, and (q,i_a) is a virtual pair.

The following table contains lists of isoclasses of virtual pairs of small cardinality.

Virtual pairs of small cardinality

Some algorithms runing on GAP

The most relevant functions are:

uncfg:

given a virtual pair [q,qi], uncfg([q,qi]) gives a record

(uncfg.equations,uncfg.generators)

where the equations are given as lists elements of the form "[[a,b],[c,d]]", meaning "ab=cd",

and generators are lists of the form

[ [[],["f",[1,1]],["g",[2,1]],["g",[2,2]],["g",[2,3]]],

[["f",[1,2]],["f",[1,3]]],

[["f",[2,2]],["f",[3,3]]],

[["g"[3,3]]] ]

meaning

1=f(1,1)=g(2,1)=g(2,2)=g(2,3),

first generator is f1:=f(1,2)=f(1,3),

second generator is f2:=f(2,2)=f(3,3),

third generator is f3:=g(3,3)

This gives simultaneously the generators of the group Unc and the value of the maps f,g:X\times X\to Unc

invariantVirtual:

invariantVirtual(pd,q,qi)

gives the non-commutative invariant associated to the virtual link corresponding to the planar diagram, and the universal non-commutative 2-cocycle pair f,g:X\times X\to Unc.

The invariant is given as a list with the colorings together with the corresponding element of the group Unc, written in terms of the generators f1, f2, ..., but without any reduction that could be eventualy be performed using the equations.

A planar diagram is a list of crossings with signs, with the same sintaxis as for classical crossings, and with "0" (instead of +1 or -1) for the "sign" if the crossing is virtual. For example, for this Kishino knot:

its planar diagram is [[-1,[4,1,5,2]], [0,[5,3,6,2]], [1, [3,7,4,6]], [-1, [10, 7, 11, 8]], [0, [11,9,12,8]], [1,[9,1,10,12]]].