Involutory biquandles and singular knots and links

Abstract: We de ne a new algebraic structure for singular knots and links. It extends the notion of a bikei (or involutory biquandle) from regular knots and links to singular knots and links. We call this structure a singbikei. This structure results from the generalized Reidemeister moves representing singular isotopy. We give several examples on singbikei and we use singbikei to distinguish several singular knots and links.


Introduction
Singular knots and links are viewed diagrammatically as knots and links with some of the crossings being 4valent rigid vertices. The theory of Vassiliev Invariants of knots and links shed the light on studying singular knots and links as a larger space that involves usual knots and links as a subspace. See [1 -3]. Since then, many knot and link invariants have been generalized to singular knots and links. For example see [4 -8].
Kei and quandles are algebraic structures, which were constructed to describe knots and links via generators and relations resulting from the arcs and the crossings of a knot or link diagram, and respecting the invariance of these diagrams under the Reidemeister moves. See [9 -17].
Kei and quandles for singular knots were constructed in [18,19], respectively. This paper introduces a new algebraic structure that generalizes involutory biquandles to singular knots. We call this structure singbikei. We give a plethora of non-trivial natural algebraic objects in examples that satisfy the axioms of this new algebraic structure. As a byproduct of this structure, we show how to apply the theory for distinguishing singular knots by giving several examples. This paper is organized as follow. In Section 2 we give the basic concepts and terminology for kei and bikei. We also de ne singular knots and links and their isotopy invariance. In Section 3 we introduce the structure of a bikei for singular knots and links and we give several examples on this new structure with some related results. In Section 4 we give some examples of singular knots and links and distinguish them using several singular bikei colorings.
We begin this section with the de nition of a kei, and after the de nition we will see how the axioms of a kei result from the three Reidemeister moves. The coloring of a regular crossing is drawn as in the following gure. De nition 2.1. A kei (involutory quandle) is a set X with a binary operation ▷: X×X → X satifying the following three axioms: The three axioms of a kei result from the invariance of the three Reidemeister moves as in the following gures. Now, instead of one operation at a crossing, two operations are de ned, and instead of coloring the arcs from the top, the arcs are colored from left to right, so we get: Next we give the de nition of a bikei. After this we will see how the axioms of a bikei result from the three Reidemeister moves.

De nition 2.2.
A bikei (involutory biquandle) is a set X with two binary operations * , * :X × X → X such that for all x, y, z ∈ X, we have The axioms of a bikei result from the invariance of the three Reidemeister moves as in gures 4, 6 and 7.  The Reidemeister move RII means that the over crossing and under crossing operations do not depend on which way the crossing is rotated.

Fig. 6. Rotated coloring crossing
The Reidemeister move RIII gives us what is called the exchange laws: Let X and Y be bikei with operations * X , * X and * Y , A bikei isomorphism is a bijective bikei homomorphism, and two bikei are isomorphic if there is a bikei isomorphism between them. Typical examples of bikei include the following : -A non-empty set X with operation x * y = x * y = σ(x), where σ is any involution from X to X, is a bikei. It is called a constant action bikei.
, then any Λ -module X with x * y = tx + (s − t)y, and x * y = sx, is called an Alexander bikei.
-A group X = G with x * y = yx − y and x * y = x is a bikei. It is called the core bikei of the group G.
A singular link in S is the image of a smooth immersion of n circles in S that has nitely many double points, called singular points. Two singular knots K and K are isotopy equivalent if we can get one of them from the other by a nite sequence of the generalized Reidemeister moves RI, RII, RIII, RIVa, RIVb and RV in the following gure.

Construction of singbikei
We will de ne the notion of a singbikei and give some examples and use them to construct an invariant of singular knots and links. The invariant is the set of colorings of a given singular knot or link by a singbikei. We draw the colorings of the regular and singular crossings as in the following gure.

The colorings of a regular crossing
The colorings of a singular crossing Since our singular crossings are unoriented, we need the operations to be symmetric in the sense that if we rotate the crossing in the right diagram of the above gure by , or degrees, the operations should stay the same in order for colorings to be well-de ned. Therefore we get the following three axioms: We have 5 generalized Reidemeister moves; I, II, III are for regular crossings; IV, V are for singular crossings. Next we show how the generalized Reidemeister moves induce relations considering the colorings of the singular crossings. The following de nition is coming from the generalized Reidemeister moves and the axioms are justi ed from the gures [9][10][11].
De nition 3.1. Let (X, * , * ) be a bikei. Let R and R be two maps from X × X to X. Then (X, * , * , R , R ) is called a singbikei if, in addition to the three axioms 3.1, 3.2 and 3.3, the following axioms are satis ed R (x * y, z * y) = R (x, z) * (y * z) The following straightforward lemma makes the set of colorings of a singular knot or link by a singbikei an invariant of singular knots and links.

Lemma 3.2. The set of colorings of a singular knot by a singbikei does not change by the Reidemeister moves RI, RII, RIII, RIVa, RIVb and RV.
Example 3.3. It is known that every kei is a bikei with the operations x * y = x ▷ y and x * y = x. Then X is a bikei with these operations so (X, * , * , R , R ) is a singbikei if R , R ∶ X × X → X satisfy the following equations: Example 3.4. Let X be a set and σ ∶ X → X be any involution on X, (i.e any map such that σ = Id X ) with x * y = x * y = σ(x). Let R , R : X × X → X be two maps, then (X, * , * , R , R ) is a singbikei if R and R satisfy the following equations: Proposition 3.5. Let X = Z n and σ ∶ Z n → Z n be given by one of the following rules This completes the proof.
Sometimes these are the only linear (i.e functions of the form f (x) = ax+b ) involutions σ on Z n and sometimes Z n has other linear involutions. For example if X = Z , in addition to the previous solutions, are also linear involutions in Z .
Lemma 3.6. If n is prime, then the only linear formulas for an involution σ on Z n are: Since σ is an involution, we have Since c = in Z n and n is prime, we have -If d = , then σ(x) = x.
-If d = n , then n is even and prime. Therefore n = so d = , then σ(x) = x + , which is included in the case when c = n − .
Theorem 3.7. Let X = Z n and σ ∶ Z n → Z n be an involution and x * y = x * y = σ(x). Then R , R : Z n ×Z n → Z n given below make (Z n , * , * , R , R ) a singbikei. Proof. We show that R and R satisfy all the equations in Example 3.4,     (1-s). Let R , R : X × X → X be two maps. Then any Λ -module X with x * y = tx + (s − t)y, and x * y = sx, is a singbikei if R and R satisfy the following equations:

Example 3.9. Let Λ = Z[t, s] (t − , s − , (s−t)( −s)) be the quotient of the ring of two-variable polynomials with integer coe cients such that s = t = by the ideal generated by (s-t)
We call such a singbikei an Alexander singbikeis. For example, 1. Z with x * y = x + y and x * y = x, R (x, y) = x + y and R (x, y) = x + y satisfy all the equations in Example 3.9. So (Z , * , * , R , R ) is an Alexander singbikei. 2. Z with x * y = x + y and x * y = x, R (x, y) = x + y and R (x, y) = x + y satisfy all the equations in Example 3.9. So (Z , * , * , R , R ) is an Alexander singbikei.
Example 3.10. Let X = G be a group with x * y = yx − y and x * y = x. Let R , R : X × X → X be two maps, then (X, * , * , R , R ) is a singbikei if R and R satisfy the following equations: x = R (y, R (x, y)) = R (R (x, y), R (x, y)) y = R (R (x, y), x) = R (R (x, y), R (x, y)) R(x, y) = (R (y, R (x, y)), R (R (x, y), x)) Proof. We show that R and R satisfy the equations in Example 3.10.
This completes the proof.

Applications
In this section we use singbikei and coloring invariants to distinguish singular knots and links.  In Figure 12 (a), the relations at the crossings give In Figure 12 (b), the relations at the crossings give Thus the set of colorings is The solution set is the same for both of the sets of colorings above. Therefore, this coloring invariant fails to distinguish these two singular knots. (b) Let X = G be a group generated by (yx − ) = with x * y = yx − y and x * y = x, R (x, y) = (xy − ) x and R (x, y) = xy − x.  In Figure 13 (a), the relations at the crossings give Thus the set of colorings is {(x, y, z) ∈ G × G × G ∶ = (xy − ) , (xy − ) = (yz − ) }. In Figure 13 (b), the relations at the crossings give Thus the set of colorings is {(x, y, z One can always choose a group G such that these two coloring sets are distinct.  In Figure 14 (a), the relations at the crossings give Thus the set of colorings is {(x, y, z) ∈ G × G × G ∶ x = y = z}. In Figure 14 (b), the relations at the crossings give Thus the set of colorings is One can always choose a group G such that these two coloring sets are distinct.

Example 4.3.
Consider the two singular links in the graph below. Each of them has one singular crossing followed by (n + ) regular crossings, let X = G be a group generated by (yx − ) = with x * y = yx − y and x * y = x, R (x, y) = (xy − ) x and R (x, y) = xy − x. In Figure 15    In Figure 16  In Figure 16 (b), the relations at the crossings give Thus the set of colorings is {(x, y, z) ∈ G × G × G ∶ z = yx − y}. Therefore, this coloring invariant distinguishes these two singular knots.
Example 4.5. Consider the two singular knots in the graph below, let X = G be a group generated by (yx − ) = with x * y = yx − y and x * y = x, R (x, y) = (xy − ) x and R (x, y) = xy − x. In Figure 17  Thus the set of colorings is G × G.
In Figure 17 (b), the set of colorings is G. Therefore, this coloring invariant distinguishes these two singular knots.