Biquandle

In mathematics, biquandles and biracks are generalizations of quandles and racks. Whereas the hinterland of quandles and racks is the theory of classical knots, that of the bi-versions, is the theory of virtual knots. In mathematics, biquandles and biracks are generalizations of quandles and racks. Whereas the hinterland of quandles and racks is the theory of classical knots, that of the bi-versions, is the theory of virtual knots. Biquandles and biracks have two binary operations on a set X {displaystyle X} written a b {displaystyle a^{b}} and a b {displaystyle a_{b}} . These satisfy the following three axioms: 1. a b c b = a c b c {displaystyle a^{bc_{b}}={a^{c}}^{b^{c}}} 2. a b c b = a c b c {displaystyle {a_{b}}_{c_{b}}={a_{c}}_{b^{c}}} 3. a b c b = a c b c {displaystyle {a_{b}}^{c_{b}}={a^{c}}_{b^{c}}}