In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots, the place that quandles occupy in the theory of classical knots. Biracks and racks have the same relation, while a biquandle is a birack which satisfies some additional conditions.
Biquandles and biracks have two binary operations on a set
X
ab
ab
1.
(ab)
cb | |
={ac}
bc | |
2.
{ab}
cb |
={ac}
bc |
3.
cb | |
{a | |
b} |
=
c} | |
{a | |
bc |
These identities appeared in 1992 in reference [FRS] where the object was called a species.
The superscript and subscript notation is useful here because it dispenses with the need for brackets. For example,if we write
a*b
ab
an{**}b
ab
1.
(an{**}b)n{**}(c*b)=(an{**}c)n{**}(bn{**}c)
2.
(a*b)*(c*b)=(a*c)*(bn{**}c)
3.
(a*b)n{**}(c*b)=(an{**}c)*(bn{**}c)
If in addition the two operations are invertible, that is given
a,b
X
x,y
X
xb=a
yb=a
X
For example, if
X
ab
ab=a
For a birack the function
S:X2 → X2
b). | |
S(a,b | |
a)=(b,a |
Then
1.
S
2.
S1S2S1=S2S1S2
In the second condition,
S1
S2
S1(a,b,c)=(S(a,b),c)
S2(a,b,c)=(a,S(b,c))
To see that 1. is true note that
S'
b)=(a,b | |
S'(b,a | |
a) |
is the inverse to
S
To see that 2. is true let us follow the progress of the triple
(c,bc,a
bcb |
)
S1S2S1
(c,bc,a
bcb |
)\to
b,a | |
(b,c | |
bcb |
)\to
bab | |
(b,a | |
b,c |
)\to(a,ba,
bab | |
c |
).
On the other hand,
(c,bc,a
bcb |
)=(c,bc,
a | |
cbc |
)
S2S1S2
(c,bc,
a | |
cbc |
)\to(c,ac,
ac | |
{b | |
c} |
)\to(a,ca,
ac | |
{b | |
c} |
)=(a,ca,
a} | |
{b | |
ca |
)\to(a,ba,
c | |
aba |
)=(a,ba,
bab | |
c |
).
Any
S
Examples of switches are the identity, the twist
T(a,b)=(b,a)
S(a,b)=(b,ab)
ab
A switch will define a birack if the operations are invertible. Note that the identity switch does not do this.
A biquandle is a birack which satisfies some additional structure, as described by Nelson and Rische.[1] The axioms of a biquandle are "minimal" in the sense that they are the weakest restrictions that can be placed on the two binary operations while making the biquandle of a virtual knot invariant under Reidemeister moves.