2.1 A Brief Review of Virtual Knots

Let $\Sigma _g$ be a closed orientable surface with genus g and let K be an embedded circle in $\Sigma _g\times [0, 1]$ . Assume we have another embedded circle $K'\subset \Sigma _{g'}\times [0, 1]$ , we say K and $K'$ are stably equivalent if one can be obtained from the other by isotopy in the thickened surfaces, homeomorphisms of the surfaces, and addition or subtraction of empty handles. We define the virtual knots to be the stable equivalence classes of circles embedded in thickened surfaces. For a virtual knot K, the minimal genus of the surface $\Sigma _g$ is called the supporting genus of K. By using some classical technique in 3-manifold topology, Kuperberg [Reference Kuperberg34] proved that the embedding of a virtual knot in the minimal supporting genus thickened surface is unique. It follows that if two classical knots are stably equivalent as virtual knots, then they are also equivalent as classical knots. This implies the virtual knot theory is indeed an extension of the classical knot theory.

From the diagrammatic viewpoint, a virtual knot can be interpreted by virtual knot diagrams. A virtual knot diagram is an immersed circle in the plane with finitely many double points. By replacing each double point with an overcrossing, or an undercrossing, a virtual crossing (denoted by a small cricle), we obtain a virtual knot diagram. Obviously if there exists no virtual crossing the virtual knot diagram represents a classical knot. We say a pair of virtual knot diagrams are equivalent if they can be connected by a sequence of generalized Reidemeister moves, see Figure 1. Now we can define virtual knots as the equivalence classes of virtual knot diagrams up to generalized Reidemeister moves.

Figure 1:

Generalized Reidemeister moves.

The two definitions above are closely related. Assume we have an embedded circle in a thickened surface; now consider a projection of the circle to the plane in general position. If the preimage of a double point is an overcrossing (resp.˜undercrossing) in the thickened surface, then we still use an overcrossing (resp.˜undercrossing) to denote it. If the two strands of the preimage of a double point locate in two different levels, then we use a virtual crossing to denote it. In other words, the virtual crossings can be regarded as artifacts of the projection of the surface to the plane. The readers are referred to [Reference Kauffman29] for more details. Conversely, suppose we have a virtual knot diagram K on the plane. By taking one-point compactification of the plane we obtain a virtual knot diagram on $S^2$ . For each virtual crossing, we add a 2-handle locally to eliminate the crossing. Finally, we obtain an embedded circle in $\Sigma _{c_v(K)}\times [0, 1]$ , where $c_v(K)$ denotes the number of virtual crossings in K. The following theorem shows that the two definitions above are equivalent.

Another way to understand virtual knots is to regard them as Gauss diagrams. Let K be a virtual knot diagram that can be seen as an immersed circle in the plane. Consider the preimage of this immersed circle with an anticlockwise orientation. For each real crossing point, we draw a chord directed from the preimage of the overcrossing to the preimage of the undercrossing. Finally, we assign a sign to each chord according to the sign of the corresponding crossing point. We call this chord diagram the Gauss diagram of K and use $G(K)$ to denote it; see Figure 2 for a simple example. We note that all virtual crossing points are ignored on the Gauss diagram.

Figure 2:

Virtual trefoil knot and its Gauss diagram.

Although there may exist infinitely many different virtual knot diagrams that correspond to the same Gauss diagram, we have the following correspondence between them.

Many virtual knot invariants have been discovered since the introduction of virtual knot theory. Several classical knot invariants can be directly extended to the virtual world. For example, the knot group, the knot quandle and the Jones polynomial can be similarly defined for virtual knots [Reference Kauffman29]. Some generalizations of the Alexander polynomial for virtual knots can be found in [Reference Sawollek43, Reference Silver and Williams45], and some generalizations of the Jones polynomial can be found in [Reference Dye and Kauffman13, Reference Miyazawa37, Reference Miyazawa38]. Readers should refer to [Reference Fenn, llyutko, Kauffman and Manturov14] for some recent progress and open problems in virtual knot theory.

2.2 Chord Index

Let K be a virtual knot diagram and let $G(K)$ be the corresponding Gauss diagram. According to the one to one correspondence between the real crossing points in K and chords in $G(K)$ , we will use the same symbol to denote a real crossing in K and its corresponding chord in $G(K)$ . Choose a chord c in $G(K)$ ; we associate four integers to c as follows:

1. (a) $r_+(c)=$ the number of positive chords crossing c from left to right;

2. (b) $r_-(c)=$ the number of negative chords crossing c from left to right;

3. (c) $l_+(c)=$ the number of positive chords crossing c from right to left;

4. (d) $l_-(c)=$ the number of negative chords crossing c from right to left.

See Figure 3. Now we define the index of c as

$$ \begin{align*} \text{Ind}(c)=r_+(c)-r_-(c)-l_+(c)+l_-(c). \end{align*} $$

Figure 3:

The definition of the chord index.

Now we follow [Reference Folwaczny and Kauffman16] to give another definition of the chord index from the viewpoint of knot diagrams. Before proceeding to give the definition, we need to take a quick review of the linking number in virtual knot theory. Let $L=K_1\cup K_2$ be a 2-component virtual link diagram. We use Over $(C)$ (resp. Under $(C)$ ) to denote the set of crossings between $K_1$ and $K_2$ that we encounter as overcrossings (resp. undercrossings) when we travel along $K_1$ . Now we define the over linking number $lk_O(L)=\sum _{c\in \text {Over}(C)}w(c)$ and the under linking number $lk_U(L)=\sum _{c\in \text {Under}(C)}w(c)$ , where $w(c)$ is the writhe of c. Note that if L is classical, we always have $lk_O(L)=lk_U(L)$ . But when L has some virtual crossings, this is not true in general.

We turn to the definition of chord index using over linking number and under linking number. Let K be a virtual knot diagram and let c be a real crossing point of it. By smoothing c along the orientation of $K,$ we obtain a 2-component link $L=K_1\cup K_2$ , where the order of $K_1$ and $K_2$ is indicated in Figure 4. The index of the crossing point c can be defined as:

$$ \begin{align*} \text{Ind}(c)=lk_O(L)-lk_U(L). \end{align*} $$

Figure 4:

Smooth the crossing point c.

We end this section with some useful properties of the chord index. The details of the proof can be found in, for example [Reference Cheng and Gao8, Reference Folwaczny and Kauffman16, Reference Kauffman31].

2.3 Writhe Polynomial

Now we explain how to use the chord index to define a polynomial invariant of virtual knots. As before, we use K to denote a virtual knot diagram and c to denote a real crossing of K. Proposition 2.3 allows us to consider the following integers:

$$ \begin{align*} a_n(K)= \begin{cases} \sum_{\text{Ind}(c)=n}w(c)& \text{if } n\neq0,\\ -\sum_{n\neq 0}a_n(K)& \text{if } n=0, \end{cases} \end{align*} $$

where $w(c)$ denotes the sign of c. We note that $a_0$ can be also interpreted as $\sum _{\mathrm{Ind}(c)=0}w(c)-w(K)$ , here $w(K)$ denotes the writhe of K. The following theorem can be easily derived from Proposition 2.3.

For convenience, we rewrite these invariants $\{a_n(K)\}$ in the form of a polynomial. We define the writhe polynomial, which was introduced in [Reference Cheng and Gao8], as

$$ \begin{align*} W_K(t)=\sum_{n\neq0}a_n(K)t^n. \end{align*} $$

The writhe polynomial $W_K(t)$ has many applications in virtual knot theory. First, $W_K(t)=0$ if K is classical, since all crossing points have index zero in this case. Hence, whenever $W_K(t)\neq 0$ , K must be non-classical. On the other hand, the writhe polynomial is quite sensitive to some symmetries of virtual knots. For example, consider the virtual trefoil knot in Figure 2. Direct calculation shows that its writhe polynomial is $t+t^{-1}$ ; however, the mirror image of it has writhe polynomial $-t-t^{-1}$ . Some examples in [Reference Cheng and Gao8] show that writhe polynomial can also be used to distinguish some virtual knots from their inverses. According to the definition of the chord index, it is evident that $|a_n(K)| (n\neq 0)$ gives a lower bound for the number of crossings with index n. However, if a crossing point has index zero, then it has no contribution to the writhe polynomial. Therefore, in general we cannot obtain any information about the number of crossing points with index zero from the writhe polynomial. For example, the index of any crossing in a classical knot diagram is zero but the writhe polynomial is also zero. One approach to overcome this problem was given recently in [Reference Cheng7]. Before ending this section, we want to remark that the writhe polynomial provides not only a lower bound for the classical crossing number, but also a lower bound for the virtual crossing number [Reference Satoh and Taniguchi42]. Recently, a necessary and sufficient condition for two virtual knots having the same writhe polynomial was given by T. Nakamura, Y. Nakanishi, and S. Satoh in [Reference Nakamura, Nakanishi and Satoh40].

3.1 A Quick Review of the Quandle Structure

To set the stage, we recall some basic notions of quandle and its generalizations.

The notion of quandle was defined by Joyce in [Reference Joyce21] (and independently by Matveev in [Reference Matveev36] with the name distributive groupoid). Given a classical knot diagram K, the knot quandle (also called fundamental quandle) $Q(K)$ is generated by the arcs of K and subject to the relations indicated in Figure 5. It is well known that the knot quandle distinguishes knots up to mirrors and reverses. However, similar to the knot group, in general, the knot quandle is difficult to deal with. In practise, a more easily computable invariant is the number of quandle homomorphisms from $Q(K)$ to a fixed finite quandle Q. We call it the coloring invariant and use $\text {Col}_Q(K)$ to denote it. From the viewpoint of knot diagram, a coloring assigns to each arc of the knot diagram an element of Q such that at each crossing the coloring rule indicated in Figure 5 is satisfied.

Figure 5:

The coloring rule at each crossing.

The knot quandle can be similarly defined for virtual knots. For a given virtual knot diagram, corresponding to each segment with breaks at under-crossings take a generator and corresponding to each classical crossing point take a relator as indicated in Figure 5. The reader is reminded that each segment traverses many (or maybe none) under-crossings and virtual crossings, and each virtual crossing provides no relator. However, the knot quandle is not good at distinguishing virtual knots. For example, the knot quandle of the virtual trefoil described in Figure 2 is trivial. For this reason, R. Fenn, M. Jordan-Santana and L. H. Kauffman [Reference Fenn, Jordan-Santana and Kauffman15] introduced a more general algebraic structure, say the biquandle. Note that the definition used here was suggested in [Reference Kaestner, Nelson and Selker24], which is a bit different from the original definition introduced in [Reference Fenn, Jordan-Santana and Kauffman15]. But it is easy to see that they are essentially the same.

The following is a simple observation.

Proof Taking $z=y$ , now the first condition in the third axiom of biquandle becomes

$$ \begin{align*} (y\circ y)\circ (x\ast y)=(y\circ x)\circ (y\circ x). \end{align*} $$

Since $x\ast y=y\circ x$ , it follows that

$$ \begin{align*} (y\circ y)\circ (y\circ x)=(y\circ x)\circ (y\circ x). \end{align*} $$

Recall that $-\circ (y\circ x)$ is invertible; hence, we obtain $y\circ y=y\circ x=x\ast y$ . Together with the first axiom $y\circ y=y\ast y$ , we have $y\ast y=x\ast y$ . The desired result follows directly, because $-\ast y$ is invertible. ▪

Later the biquandle structure was extended by Kauffman and Manturov to the virtual biquandle [Reference Kauffman and Manturov32] by adding a relation at each virtual crossing point. The readers are referred to [Reference Carter2] for more details of quandle ideas. We will come back to the biquandle structure in Section 4.

3.2 Indexed Quandle

The parity biquandle was introduced in [Reference Kaestner and Kauffman23] and studied in [Reference Kaestner, Nelson and Selker24], where the parity was used to generalize the quandle structure. What we want to do is replace the parity with chord index.

We remark that each indexed quandle $(X, \ast _i)$ includes a quandle $(X, \ast _0)$ . The following are some examples of the indexed quandles.

• • Let $(Q, \ast )$ be a quandle; then Q can be thought of as an indexed quandle if we define $\ast _i=\ast $ for any $i\in \mathbb {Z}$ .

• • Let $(Q, \ast )$ be a quandle; another way to regard Q as an indexed quandle is introducing $\ast _0=\ast $ and $a\ast _{i} b=a (i\neq 0)$ for any $a, b\in Q$ .

• • Let $X=\mathbb {Z}[t, t^{-1}]$ be the Laurent polynomial ring. One defines $a\ast _i b=ta+(1-t)b+i$ for any $a, b\in \mathbb {Z}[t, t^{-1}]$ . Then $(X, \ast _i)$ is an indexed quandle.

• • Let G be a group. One defines $a\ast _i b=\phi (ab^{-1})bz^i$ for any $\phi \in \text {Aut}(G)$ and $z\in Z(G)$ (the center of G). Then $(G, \ast _i)$ is an indexed quandle.

For a given virtual knot diagram K, we define the indexed knot quandle $I(K)$ to be the indexed quandle generated by the arcs of K, and each real crossing point gives a relation that depends on the index; see Figure 6.

Figure 6:

The indexed coloring rule at each crossing.

Notice that when K is a classical knot diagram, the indexed knot quandle $I(K)$ reduces to the classical knot quandle $Q(K)$ .

Figure 7:

The invariance of $I_K$ under $\Omega _3$ .

For a finite indexed quandle X, we denote by $\text {Col}_X(K)$ the number of homomorphisms from the indexed knot quandle $I(K)$ to X.

Recall that any indexed quandle $(X, \ast _i)$ defines the quandle $(X, \ast _0)$ . For a classical knot K, it holds that

$$ \begin{align*} \text{Hom}(I(K), (X, \ast_i))=\text{Hom}(Q(K), (X, \ast_0)), \end{align*} $$

we have $\text {Col}_{(X, \ast _i)}(K)=\text {Col}_{(X, \ast _0)}(K)$ . However, this is not the case for virtual knots.

Example 3.7 Let K be a virtual knot diagram as shown in Figure 8. For the set $D_3=\{0, 1, 2\}$ , the operation $i\ast j=2j-i \pmod 3 $ defines the quandle $(D_3, \ast )$ , and $i\ast _k j=2j-i+k\pmod { 3}$ defines the indexed quandle $(D_3, \ast _k)$ . Then we have

$$ \begin{align*} {\mathrm{Col}}_{(\mathrm{D}_3, \ast)}{\mathrm{(K)}}={\mathrm{9}}\quad{\mathrm{and}}\quad {\mathrm{Col}}_{(\mathrm{D}_3, \ast_k)}{\mathrm{(K)=0.}} \end{align*} $$

Since the trefoil knot T satisfies $ \rm \text {Col}_{(D_3, \ast )}(T)=\text {Col}_{(D_3, \ast _k)}(T)=9$ , we have $K\neq T$ .

Figure 8:

The virtualization of the trefoil knot.

3.3 An Enhancement of the Coloring Invariant via Indexed Quandle 2-cocycles

The coloring invariant $\text {Col}_Q(K)$ has many applications in knot theory. For example, $\text {Col}_Q(K)$ can be used to provide lower bounds for braid index, tunnel number, and unknotting number [Reference Clark, Elhamdadi, Saito and Yeatman11]. However, there also exist some disadvantages to $\text {Col}_Q(K)$ . For instance, for any finite quandle Q, $\text {Col}_Q(K)$ cannot distinguish the left-hand trefoil from the right-hand trefoil, since the trefoil knot is invertible. In [Reference Carter, Jelsovsky, Kamada, Langford and Saito4], J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford, and M. Saito introduced the quandle (co)homology theory. In particular, with a given quandle 2(3)-cocycle, one can define an enhancement of the (shadow) coloring invariant, say the quandle cocycle invariant. There are many examples that show that these cocycle invariants are more powerful than the original coloring invariants. As an example, for some suitably chosen quandle and quandle 3-cocycle the cocycle invariant takes different values on the left-hand trefoil and the right-hand trefoil [Reference Rourke and Sanderson41]. Following the main idea of the quandle cocycle invariant we want to define the notion of indexed quandle 2-cocycle in this subsection. Analogous to the quandle 2-cocycle, each indexed quandle 2-cocycle also can be used to define a generalized invariant of $\text {Col}_{(Q, \ast _i)}(K)$ .

Definition 3.8 Let $(Q, \ast _i)$ be an indexed quandle and let A be an abelian group; we call a map $\phi : Q\times Q\rightarrow A$ an indexed quandle 2-cocycle if for any $a, b, c\in Q$ and $i, j\in \mathbb {Z},$ it satisfies

$$ \begin{align*} \phi(a, c)^{-1}\phi(a\ast_{i-j}b, c)\phi(a, b)\phi(a\ast_i c, b\ast_j c)^{-1}=1 \quad \text{and}\quad \phi(a, a)=1. \end{align*} $$

For a given virtual knot diagram K and a finite indexed quandle $(Q, \ast _i)$ , we choose a coloring $f: I(K)\rightarrow (Q, \ast _i)$ . For a crossing x with index i, if the three arcs around x have colors $a, b,$ and $a\ast _i b$ (see Figure 6), then we associate a weight $B(x, f)=\phi (a, b)^{w(x)}$ with the crossing point x. As before, $w(x)$ denotes the sign of x. Now we define the indexed quandle 2-cocycle invariant (associated with $\phi $ ) to be

$$ \begin{align*} \Phi _{\phi }(K)=\sum \limits _f\prod \limits _xB(x, f), \end{align*} $$

where f runs over all homomorphisms from $I_K$ to $(Q, \ast _i)$ , and x runs over all real crossing points of K. Obviously, if $\phi $ sends each element of $Q\times Q$ to the identity element, then $\Phi _{\phi }(K)$ reduces to the coloring invariant $\text {Col}_{(Q, \ast _i)}(K)$ .

We remark that the indexed quandle 2-cocycle invariant can be extended by replacing $\phi $ with a sequence of homomorphisms $\phi _i (i\in \mathbb {Z})$ . More precisely, we use $\psi $ to denote a sequence of $\{\phi _i\}_{i\in \mathbb {Z}}$ where each $\phi _i$ represents a map from $Q\times Q$ to A. We say $\psi $ is a generalized indexed quandle 2-cocycle if for any $a, b, c\in Q$ and $i, j\in \mathbb {Z},$ we have

$$ \begin{align*} \phi_i(a, c)^{-1}\phi_i(a\ast_{i-j}b, c)\phi_{i-j}(a, b)\phi_{i-j}(a\ast_i c, b\ast_j c)^{-1}=1 \quad \text{and}\quad \phi_0(a, a)=1. \end{align*} $$

With a fixed generalized indexed quandle 2-cocycle $\psi =\{\phi _i\}_{i\in \mathbb {Z}}$ and a coloring f, we associate a weight $\mathfrak {B}(x, f)=\phi _i(a, b)^{w(x)}$ with the crossing point x in Figure 6. Now we define the generalized indexed quandle 2-cocycle invariant as follows:

$$ \begin{align*} \Psi _{\psi }(K)=\sum \limits _f\prod \limits _{x}\mathfrak {B}(x, f), \end{align*} $$

where the product takes over all crossing points and the sum takes over all colorings. In the same way, one can prove that $\Psi _{\psi }(K)$ is a virtual knot invariant.

We end this section with a simple example of the generalized indexed quandle 2-cocycle invariant. Consider the indexed quandle that consists of one element a and choose the abelian group $A=\mathbb {Z}[t, t^{-1}]$ . We define a generalized indexed quandle 2-cocycle $\psi =\{\phi _i\}$ by letting $\phi _i(a, a)=t^i (i\neq 0)$ and $\phi _0(a, a)=0$ . Note that there exists only one coloring. Now the generalized indexed quandle 2-cocycle invariant can be read as

$$ \begin{align*} \Psi _{\psi }(K)=\sum _{\mathrm{Ind}(x)\neq 0}w(x)t^{\mathrm{Ind}(x)}=W_K(t), \end{align*} $$

which is exactly the writhe polynomial we discussed in Section 2. It means that the writhe polynomial can be understood as a special case of the generalized indexed quandle cocycle invariant.

3.4 Abelian Extensions of Indexed Quandle by 2-cocycles

In group theory, it is well known that there is a one to one correspondence between the set of isomorphism classes of central extensions of G by A and the cohomology group $H^2(G, A)$ . The analogous relation between quandle extensions and quandle 2-cocycles was given by J. S. Carter et al in [Reference Carter, Elhamdadi, Nikiforou and Saito3]. In this subsection, we want to explore the relation between the extensions of indexed quandles and the generalized indexed quandle 2-cocycles.

For an indexed quandle $(Q, \ast _i)$ , an abelian group $A,$ and a sequence of maps $\psi =\{\phi _i:Q\times Q\rightarrow A, i\in \mathbb {Z}\}$ , we define a set $E(Q, A, \psi )=A\times Q$ equipped with a sequence of binary operations

$$ \begin{align*} (a_1, x_1)\ast _i(a_2, x_2)=(a_1\phi _i(x_1, x_2), x_1\ast _ix_2). \end{align*} $$

The following proposition says $E(Q, A, \psi )$ is an indexed quandle if and only if $\psi $ is a generalized indexed quandle 2-cocycle. In this case, we say the set $E(Q, A, \psi )$ is an abelian extension of $(Q, \ast _i)$ .

Proof We assume $\psi $ is a generalized indexed quandle 2-cocycle. Recall that this means that for any $x_1, x_2, x_3$ of $Q,$ we have

$$ \begin{align*} \phi_i(x_1, x_3)^{-1}\phi_i(x_1\ast_{i-j}x_2, x_3)\phi_{i-j}(x_1, x_2)\phi_{i-j}(x_1\ast_i x_3, x_2\ast_j x_3)^{-1}&=1, \\ \phi_0(x_1, x_1)&=1. \end{align*} $$

First, note that

$$ \begin{align*} (a_1, x_1)\ast_0(a_1, x_1)&=(a_1\phi_0(x_1, x_1), x_1\ast_0x_1)=(a_1, x_1), \\ (a_3, x_3)\ast_i^{-1}(a_2, x_2)&=(a_3(\phi_i(x_3\ast_i^{-1}x_2, x_2))^{-1}, x_3\ast_i^{-1}x_2). \end{align*} $$

Next, it suffices to prove

$$ \begin{align*} ((a_1, x_1)\ast_i(a_2, x_2))\ast_j(a_3, x_3)=((a_1, x_1)\ast_j(a_3, x_3))\ast_i((a_2, x_2)\ast_{j-i}(a_3, x_3)). \end{align*} $$

One computes

$$ \begin{align*} &((a_1, x_1)\ast_i(a_2, x_2))\ast_j(a_3, x_3)\\ &= (a_1\phi_i(x_1, x_2), x_1\ast_ix_2))\ast_j(a_3, x_3)\\ &= (a_1\phi_i(x_1, x_2)\phi_j(x_1\ast_ix_2, x_3), (x_1\ast_ix_2)\ast_jx_3)\\ &= (a_1\phi_j(x_1, x_3)\phi_i(x_1\ast_jx_3, x_2\ast_{j-i}x_3), (x_1\ast_jx_3)\ast_i(x_2\ast_{j-i}x_3))\\ &= (a_1\phi_j(x_1, x_3), x_1\ast_jx_3)\ast_i(a_2\phi_{j-i}(x_2, x_3), x_2\ast_{j-i}x_3)\\ &= ((a_1, x_1)\ast_j(a_3, x_3))\ast_i((a_2, x_2)\ast_{j-i}(a_3, x_3)). \end{align*} $$

Conversely, if $E(Q, A, \psi )$ is an indexed quandle, one can similarly prove that $\psi $ satisfies the 2-cocycle conditions. ▪

5.1 Twisted Knot Theory and Twisted Biquandle

The end of this paper is concerned with the chord index in twisted knot theory. Recall that virtual knot theory studies the embeddings of $S^1$ in thickened closed orientable surfaces; if we do not require that the surface must be orientable, then we encounter the twisted knot theory. Twisted knot theory was first proposed by Bourgoin [Reference Bourgoin1]. A twisted knot is a stable equivalence class of $S^1$ in oriented 3-manifolds that are I-bundles over closed but not necessarily orientable surfaces. One main result in [Reference Bourgoin1] generalizes Kuperberg’s result [Reference Kuperberg34] from orientable surfaces to nonorientable surfaces. More precisely, Bourgoin mimicked Kuperberg’s approach to prove that the irreducible representative of a twisted knot is unique. Therefore, twisted knot theory is a proper extension of virtual knot theory.

One can also use twisted knot diagrams to illustrate twisted knots. A twisted knot diagram is a virtual knot diagram with some bars on edges. We say two twisted knot diagrams are equivalent if they are related by a sequence of generalized Reidemeister moves (see Figure 1) and twisted Reidemeister moves (see Figure 12). Similar to virtual knots, from each twisted knot diagram one can obtain an embedding of $S^1$ in a thickened surface, where each bar corresponds to a half-twist. It was proved in [Reference Bourgoin1] that these two definitions of twisted knots are equivalent.

Figure 12:

Twisted Reidemeister moves.

The twisted knot group and the twisted Jones polynomial were defined in [Reference Bourgoin1]. Later N. Kamada generalized the arrow/Miyazawa polynomial to twisted knots [Reference Kamada26]. Using the similar idea of last section we want to introduce the chord index of twisted knots with a twisted biquandle. Note that the twisted quandle has been introduced by N. Kamada in [Reference Kamada26], which was motivated by the twisted group defined by Bourgoin. Here the twisted biquandle discussed below cannot be thought of as a biquandle version of the twisted quandle simply, although they reduce to biquandle and quandle respectively when the knot diagram contains no bar.

With a given finite twisted biquandle $(BQ, \ast , \circ , f)$ , we can define a coloring invariant $\text {Col}_{BQ}(K)$ for each twisted knot K as follows. Choose a twisted knot diagram of K; for simplicity, we still use K to denote it. Assume there are $c_r(K)$ real crossing points and b bars in K, now these crossings and bars split K into $2c_r(K)+b$ segments. For each segment, we label an element of $BQ$ to it such that the coloring rules described in Figure 9 are satisfied. In addition, if two segments are adjacent to the same bar, then the elements on them differ by f.

Figure 13:

The invariance of $\text {Col}_{BQ}(K)$ under $\Omega _3^t$ .

In next subsection, we will focus on a special twisted biquandle $(\mathbb {Z}, a\ast b=a\circ b=a+1, f(a)=-a)$ . We will show how to associate an index for each real crossing point via this twisted biquandle.

5.2 A Concrete Example of Chord Index for Twisted Knots

We consider the colorings of twisted knots using twisted biquandle $(\mathbb {Z}, a\ast b=a\circ b=a+1, f(a)=-a)$ . A naive observation is the parity of the number of bars is preserved under twisted Reidemeister moves. For example, if a twisted knot diagram contains an odd number of bars then it can not represent a virtual knot. We continue our discussion in two cases.

First, let us consider a twisted knot K which has an odd number of bars. In this case, we have the following result.

Proof Assume K has $2n-1$ bars, denoted by $b_1, \dots , b_{2n-1}$ . We use $e_1, \cdots , e_{2n-1}$ to denote the edges of $K-\{b_1, \cdots , b_{2n-1}\}$ , where the order of $e_1, \cdots , e_{2n-1}$ is consistent with the orientation of K. Consider an edge $e_i$ , we use $o_+(e_i), o_-(e_i), u_+(e_i), u_-(e_i)$ to denote the number of positive overcrossings, negative overcrossings, positive undercrossings, and negative undercrossings on $e_i,$ respectively. Then we assign an integer $s(e_i)=u_+(e_i)+o_-(e_i)-u_-(e_i)-o_+(e_i)$ to $e_i$ .

Recall that by a coloring, we mean an assignment of integers to the segments that are obtained from K by deleting all real crossing points and bars. Along the direction of K, denote the segment adjoint to $b_1$ by a. Notice that according to the coloring rules, when we assign an integer k to $a,$ the coloring of any other segment can be derived from the coloring on a. If a coloring is well defined, then the derived coloring on a must equal k. This can be described by the following equation:

$$ \begin{align*} \sum_{i=1}^{n-1}s(e_{2i})-\sum_{i=1}^{n}s(e_{2i-1})-k=k. \end{align*} $$

Since

$$ \begin{align*} \sum_{i=1}^{n-1}s(e_{2i})+\sum_{i=1}^{n}s(e_{2i-1})=\sum_{i=1}^{2n-1}s(e_i)=0 . \end{align*} $$

It follows that $\sum _{i=1}^{n-1}s(e_{2i})-\sum _{i=1}^{n}s(e_{2i-1})$ is even, which means that k has a unique solution. ▪

Now we know that for any twisted knot K there is a unique coloring by using the twisted biquandle $(\mathbb {Z}, a\ast b=a\circ b=a+1, f(a)=-a)$ . If the colors on the four segments around a crossing c are depicted as that in Figure 9, then we define the index of c to be $\mathrm{Ind}(c)=y-x$ . As before, we define

$$ \begin{align*} a_n(K)= \begin{cases} \sum_{\text{Ind}(c)=n}w(c)& \text{if } n\neq0,\\ \sum_{\text{Ind}(c)=n}w(c)-w(K)& \text{if } n=0. \end{cases} \end{align*} $$

The following theorem is an analogue of Theorem 2.4 for twisted knots with an odd number of bars.

Now we turn to the twisted knots with an even number of bars. Note that this set contains all virtual knots, and hence all classical knots. Let K be a twisted knot with $2n$ bars, denoted by $b_1, \dots , b_{2n}$ . These bars divides the knot diagram K into $2n$ edges, say $e_1, \cdots , e_{2n}$ , where the order of $e_1, \dots , e_{2n}$ agrees with the direction of K. For each edge $e_i$ the assigned integer $s(e_i)$ can be defined as above.

According to the definition of $S(K)$ , it is evident that $S(K)=0$ if K is a virtual knot. On the other hand, since $\sum _{i=1}^ns(e_{2i-1})+\sum _{i=1}^ns(e_{2i})=0$ , we conclude that $S(K)$ is always an even integer. The relation between the existence of colorings of K and $S(K)$ is given in the following lemma.

Proof The proof is similar to the proof of Lemma 5.3. If a segment is assigned with an integer k, then the labels on other segments are determined according to the coloring rules. It is easy to find that there exists a coloring if and only if the equations

$$ \begin{align*} \sum_{i=1}^{n}s(e_{2i-1})-\sum_{i=1}^{n}s(e_{2i})+k=k \quad \text{and}\quad \sum_{i=1}^{n}s(e_{2i})-\sum_{i=1}^{n}s(e_{2i-1})+k=k \end{align*} $$

hold. Consequently, there exists a coloring of K if and only if $S(K)=0$ .▪

According to the proof above, we know that if $S(K)=0,$ then there are infinitely many different colorings, which can be obtained by coloring a fixed segment with all integers. In order to color twisted knots with nonzero $S(K)$ , we replace the twisted biquandle $(\mathbb {Z}, a\ast b=a\circ b=a+1, f(a)=-a)$ with $(\mathbb {Z}_{S(K)}, a\ast b=a\circ b=a+1, f(a)=-a)$ . In particular, if $S(K)=0,$ then we have $\mathbb {Z}_{S(K)}=\mathbb {Z}$ . For a twisted knot K, if we use the twisted biquandle $(\mathbb {Z}_{S(K)}, a\ast b=a\circ b=a+1, f(a)=-a),$ then there are exactly $S(K)$ different colorings. Fix a coloring f, for the crossing point c depicted in Figure 9 we define the index of it associated with f as $\mathrm{Ind}_f(c)=y-x \pmod {S(K)} \in \mathbb {Z}_{S(K)}$ . Obviously this definition depends on the choice of f, but what we need is an index that does not depend on the choice of colorings. Hence, it is necessary to study the indices of all colorings. Fortunately, there are only $S(K)$ different colorings totally.

Let K be a twisted knot with an even number of bars, the Gauss diagram of K can be similarly defined as the virtual knots. We still use $G(K)$ to denote it. For any chord c in $G(K)$ , it splits the circle into two semi-circles. Since there are totally an even number of bars in K, then either both semi-circles have an even number of bars, or both semi-circles have an odd number of bars. We use $C_e(K)$ to denote all the chords of the first case and $C_o(K)$ to denote all the chords of the second case.

Due to the lemma above, the set $C_o(K)$ can be divided into two pieces, say $C_o^0(K)$ and $C_o^1(K)$ , where $C_o^0(K)$ contains all the crossing points with even indices and $C_o^1(K)$ contains all the crossing points with odd indices. Since $\mathrm{Ind}_f(c)$ does not depend on the choice of f if $c\in C_e(K)$ , we can simply use $\mathrm{Ind}(c)$ to denote it. The results of the discussion above can be summarized in the form of a polynomial:

$$ \begin{align*} T_e(K)=\sum_{c\in C_o^0(K)}w(c)s_0+\sum_{c\in C_o^1(K)}w(c)s_1+\sum_{c\in C_e(K)}w(c)t^{\text{Ind}(c)}-w(K). \end{align*} $$

It is routine to check that

$$ \begin{align*} \sum_{c\in C_o^0(K)}w(c), \quad \sum_{c\in C_o^1(K)}w(c), \quad \text{and} \quad \sum_{c\in C_e(K)}w(c)t^{\text{Ind}(c)}-w(K) \end{align*} $$

are invariant under the generalized Reidemeister moves and the twisted Reidemeister moves. Therefore we omit the proof here.

The first index type invariant of twisted knots was introduced by N. Kamada in [Reference Kamada27]. In [Reference Kamada27], N. Kamada defined two polynomials of twisted knots, denoted by $\overline {Q}_K$ and $\tilde {Q}_K$ , where $\tilde {Q}_K$ is a refinement of $\overline {Q}_K$ . We remark that some of our results, for instance $\sum _{c\in C_o^0(K)}w(c)s_0$ and $\sum _{c\in C_o^1(K)}w(c)s_1,$ also can be found in the definition of $\overline {Q}_K$ . We end this paper with an example that explains the difference between our polynomial invariants and that introduced by N. Kamada.

Figure 14:

A twisted knot with an odd number of bars.