On Axioms of Biquandles

We prove that the two conditions from the definition of a biquandle by Fenn, Jordan-Santana, Kauffman [1] are equivalent and thus answer a question posed in the paper. We also construct a weak biquandle, which is not a biquandle. According to Fenn, Jordan-Santana and Kauffman [1], biquandles provide powerful invariants of virtual knots and links. It is thus desirable to simplify their axioms as much as possible. The aim of this very short note is to answer two questions regarding the definition of biquandles raised in [1], Section 4. For a background, please consult [1] or [2]. A pair (X,S) is called a switch, if S is a permutation of X such that (S × id)(id × S)(S × id) = (id × S)(S × id)(id × S). (†) Put S(x, y) = (x ◦ y, y ∗ x). Originally, the notation in [1] was S(x, y) = (yx, x ). We will use the infix notation in order to keep the computation below readable. Now, apply the left side of (†) to a triple (x, y, z) ∈ X; the result is the triple ((x◦y)◦((y ∗x)◦z), ((y ∗x)◦z)∗(x◦y), z ∗(y ∗x)). Similarly, the result of the right side on (x, y, z) is (x ◦ (y ◦ z), ((y ◦ z) ∗ x) ◦ (z ∗ y), (z ∗ y) ∗ ((y ◦ z) ∗ x)). Consequently, the identity (†) is equivalent to the following three switch identities in terms of the operations ◦, ∗: x ◦ (y ◦ z) = (x ◦ y) ◦ ((y ∗ x) ◦ z) (1) x ∗ (y ∗ z) = (x ∗ y) ∗ ((y ◦ x) ∗ z) (2) ((x ∗ y) ◦ z) ∗ (y ◦ x) = ((x ◦ z) ∗ y) ◦ (z ∗ x) (3) 1991 Mathematics Subject Classification. 08A62,57M27.