Many classes of links have been defined in trying to generalize the class of alternating links. In particular, Lickorish and Thistlewaite in [7] introduced the class of adequate links and Cromwell in [2] introduced the class of homogeneous links. Later, Kauffman in [4] introduced the class of alternative links. These classes of links are natural generalization of alternating links each in its own aspect.

We recall the definition of these classes for the sake of making this paper more self-contained. From now on and for the rest of this paper, we assume that a link is oriented unless mentioned otherwise.

First, we recall the definition of adequate links as in [7]. Let L be a link of a diagram D with crossings c1,c2,…,cn. A state of the diagram D is a function s:{c1,c2,…,cn}⟶{1,-1}, and the state diagram sD is the diagram obtained from the diagram D after smoothing all of its crossings according to the state s. We apply the A-smoothing to the crossing ci if s(ci)=1 or the B-smoothing if s(ci)=-1 in the case that the crossing is given according to the scheme in Figure 1 and the other way around in the other case. Note that sD consists of simple closed curves and the number of components will be denoted by |sD|.

Figure 1. The crossing c of the diagram D, the A-smoothing, and the B-smoothing, respectively.

There are two special state diagrams sAD and sBD. The first one is obtained when sA(ci)=1 and the second one is obtained when sB(ci)=-1 for all i such that 1≤i≤n. The link diagram is A-adequate if |sAD|>|sD| for every state diagram sD with ∑i=1ns(ci)=n-2 and is B-adequate if |sBD|>|sD| for every state diagram sD with ∑i=1ns(ci)=2-n. This is equivalent to say that the two arcs after smoothing each crossing in sAD or sBD belong to two different components. The link diagram is adequate if it is A- and B-adequate at the same time and the link is adequate if it admits a diagram that is adequate.

Thistlewaite in [11] proved that any adequate diagram of an adequate link has the minimal crossing number among all diagrams that represent the same link using the Kauffman polynomial. The same result is obtained using the Jones polynomial [7]. Moreover, the result of [11] implies that the minimal crossing diagram of an adequate link is also adequate.

Second, we recall the definition of homogeneous links as in [2]. Seifert in [9] defined an algorithm for constructing an orientable surface spanning the link L from its oriented diagram D. The deformation retraction of this surface is a graph G. The vertices of G correspond to the Seifert circles after collapsing them to points. The edges of G correspond to the twisted rectangles that connect the Seifert circles, and hence the crossings in D. The edges of G can be equipped with signs according to the type of the crossing given in Figure 2. The resulting graph with signs is called Seifert graph.

Figure 2. A positive crossing (Type I) and a negative crossing (Type II), respectively.

In any graph, a cut vertex v is a vertex whose removal increases the number of components. If the components of G-v are G1,G2,…,Gn, then G1∪v,G2∪v,…,Gn∪v are subgraphs of G obtained by cutting G at v. Cutting G at each of its cut vertices yields a family of subgraphs, no one of which has a cut vertex. These subgraphs are called the blocks of the graph G.

A Seifert graph is homogeneous if all of the edges in each block have the same sign. A link diagram is homogeneous if the corresponding Seifert graph is homogeneous. The link is homogeneous if it admits a diagram that is homogeneous.

Finally, we define alternative links by using enhanced checherboard graph [10, Theorem 19]. This graph Φ(D) is a signed planar digraph constructed from the link diagram D in the following way. We color the regions of the diagram into two colors black and white such that regions which share an arc have different colors. We then place a vertex in each region colored according to the color of the region. Two vertices are connected by a directed edge with a sign according to the scheme in Figure 3.

Figure 3. The edges and vertices of the graph Φ(D).

A planar signed digraph G is said to be alternative if it does not contain any walk with positive and negative edges. The link diagram is alternative if the corresponding graph Φ(D) is alternative. The link is alternative if it has a diagram that is alternative.

In this paper, we explain how to construct an infinite family of adequate, homogeneous, or alternative links from a given adequate, homogeneous or alternative link, respectively using the twisting technique. The twisting technique introduced for the first time in [1] to construct an infinite family of quasi-alternating links from a given quasi-alternating link by replacing a particular crossing by some rational tangle. This technique was later generalized in [8] by replacing the particular crossing by a block of rational tangles not just a rational tangle.

In this section, we briefly recall and review some properties of rational tangles that will be used in the rest of this paper. It is well-known that each rational tangle is characterized by the continued fraction of some rational number that is called the slope of the rational tangle. The continued fraction of the rational number βα is given by

where a1,a2,…,an are called the integer denominators of the continued fraction. This continued fraction will be denoted by the sequence of integers [a1,a2,…,an]. It is a simple fact that each rational number admits such a finite continued fraction.

A rational tangle is a proper embedding of the disjoint union of two arcs into a cube, the embedding sends the end points of the two arcs to four marked points NE, NW, SE, and SW with the symbols referring to the compass directions on the cube's boundary.

The characterization of a rational tangle by the continued fraction of its slope is given by the scheme as shown in Figure 4. The i-th box (i=1,2,…,n) in Figure 4 corresponds to |ai|-horizontal crossings of positive sign if ai is positive and i is odd or ai is negative and i is even and otherwise it corresponds to |ai|-horizontal crossings of negative sign, where the sign of the crossing is positive if the over arc of the crossing has positive slope and negative otherwise as pictured in Figure 1.

Figure 4. The presentation of a rational tangle according to the number of denominators being odd or even respectively.

The product of rational tangles is defined by placing one tangle on top of the other tangle by identifying the end points of the top tangle marked with SE, SW to the end points of bottom tangle marked with NE, and NW, respectively. Also, the sum of two rational tangles is defined by juxtaposing the first tangle next to the second one and identifying the end points of the left tangle marked with NW, and SW to the end points of the right tangle marked with NE, and SE, respectively. A block of rational tangles is a finite product or/and sum of rational tangles. For further discussion of rational tangles and their operations, we refer the reader to [5], and [6]. The following lemma lists some of the well-known properties of the continued fraction that can be found in many references for example [5, 6, 8] and will be used in the rest of this paper.

Lemma 2.1.

• (1)

  If [a1,a2,…,an] is a continued fraction of βα, then [-a1,-a2,…,-an] is a continued fraction of -βα.

• (2)

  If βα is a positive rational number, then there is a continued fraction [a1,a2,…,an] of nonnegative integers.

• (3)

  The rational tangle of corresponding continued fraction [a1,a2,…,an-1,±1] is isotopic to the rational tangle of corresponding continued fraction [a1,a2,…,an-1±1].

Proof. The second part of (3) follows as a consequence of the fact that isotopying the last crossing of the first tangle from the n-th box to the (n-1)-th box in the second tangle rotates such a crossing clockwise by 90∘ as requested.

The determinant of a link is an invariant that can be defined in terms of the number of signed spanning tress of the Tait graph. The Tait graph G(L) associated to the link diagram L is obtained from the checkerboard coloring of the regions of the link diagram. The vertices of G(L) are the shaded regions and the signed edges correspond to crossings according to the scheme in Figure 5.

Figure 5. The signs of the edges of the Tait graph G(L).

For a given link L of a diagram D with a crossing c, we let L0 and L1 be the links obtained by applying the A- and B-smoothings respectively to D at c according to the scheme in Figure 1. Without loss of generality, we can assume that the edge e corresponding to the crossing c is positive in G(L) by choosing the appropriate checkerboard coloring of the regions of L or by taking the mirror image of the link if required. It is well-known that the spanning trees of G(L) are in one-to-one correspondence of the spanning trees of G(L0) and G(L1). In particular, any spanning tree of G(L) corresponds to a spanning tree of G(L0) if it does not contain the edge e and to a spanning tree of G(L1) in the other case or vice versa.

The number of positive edges in any spanning tree T of G will be denoted by v(T) and the number of spanning trees of G with v(T)=v will be denoted by sv(L). The numbers ∑v(-1)vsv-1(L0), ∑v(-1)vsv(L1) will be denoted by x and y respectively for the following lemma whose proof can be obtained by adopting the proof of [1, Theorem 2.1] with the appropriate modifications. This lemma may be considered as a natural generalization of [8, Lemma3.1] and its proof.

Lemma 2.2. If a diagram of a link L* is obtained from a diagram, with a crossing c, of a link L by replacing the crossing c by a rational tangle of slope βα with corresponding continued fraction of denominators of the same sign as the sign of c, then we have

We recall the twisting technique that was first introduced in [1, Page 2452] and later generalized in [8]. This technique replaces a particular crossing in a link diagram by some rational tangle or a block of rational tangles in the class of quasi-alternating links.

Definition 3.1. Let c be a crossing of a link diagram. The rational tangle associated to the continued fraction [a1,a2,…,an] of denominators of the same sign extends the crossing c if each crossing in the i-th box in Figure 4 is of the same sign as the sign of c when i is odd and of the opposite sign when i is even according to the scheme in Section 2. A block of rational tangles extends the crossing c if each rational tangle in the block extends it.

The above definition can be modified to the case of oriented rational tangles as follows:

Definition 3.2. Let c be a crossing of a link diagram. The orientation of a rational tangle associated to the continued fraction [a1,a2, …,an] of denominators of the same sign extends the orientation of the crossing c if each crossing in the i-th box in Figure 4 is of the same type as the type of c for i=1,2,…,n according to the scheme in Figure 2. An oriented rational tangle extends the crossing c if it extends c with no orientation and its orientation extends the orientation of c. The block of oriented rational tangles extends the crossing c if each oriented rational tangle in the block extends it.

We point out that not all rational tangles that extend the crossing c can be equipped with an orientation that extends the orientation of c. Now we state the first main theorem in this paper:

Theorem 3.3. Let L be an adequate link with adequate diagram D. If L* is the link obtained by replacing any crossing in D by a block of rational tangles that extends it, then L* is adequate with corresponding adequate diagram D*.

Proof. Let c be any crossing of the adequate diagram D. By taking the mirror image if necessary, we may assume that sign(c)=1 and later use the fact that the diagram is adequate iff its mirror image is adequate to obtain the required result.

We prove the result first in the case when the crossing is replaced by an integer tangle that extends c. Let Dm be the link diagram obtained from D after replacing c by an integer tangle of m vertical or horizontal crossings that extends this crossing. In the special states sAD and sBD, the two arcs in smoothing any crossing belong to two different components since D is adequate. In particular, this is the case for the two arcs in smoothing c. Now it is obvious that this also holds for any crossing of Dm to obtain sADm or sBDm.

Now we prove the result when the crossing c is replaced by a rational tangle that extends c. For this end, we let [a1,a2,…,an] to be the continued fraction of some rational tangle that extends c. We use induction on the number of denominators of the continued fraction to prove our claim. The above argument implies the result for n=1. Now the rational tangle corresponding to the continued fraction [a1,a2,…,an-1+1] is isotopic to the rational tangle corresponding to the continued fraction [a1,a2,…,an-1,1] from the third part of Lemma 2.1. Now, it is not too hard to see that the diagram obtained by replacing the crossing c by the rational tangle of continued fraction [a1,a2,…,an-1+1] is adequate iff the diagram obtained by replacing the crossing c by the rational tangle of continued fraction [a1,a2,…,an-1,1] is adequate. Therefore, the result follows in this case using the induction hypothesis and the result when n=1. Finally, the result when the crossing c is replaced by a block of rational tangles is obtained by first replacing the crossing c by an integer tangle that extends c and then replacing each crossing of such tangle by a rational tangle that extends the corresponding crossing.

Corollary 3.4. For the links L and L*, we have c(L*)=c(L)+c(b)=c(D)+c(b) where c(b) is the number of crossings of the block of rational tangles that extends the crossing c in the link diagram D.

Proof. The result follows as a consequence of the fact that the adequate reduced diagram of a link has minimal crossing number in [11].

Corollary 3.5. A given adequate link yields an infinite family of distinct adequate links.

Proof. The set of links {L*} is infinite as it contains an infinite subset of links of distinct crossing number.

Now if we let M〈D〉 and m〈D〉 denote the maximum and minimum powers of A that occur in the Kauffman bracket of the adequate link diagram D, then in the following proposition we can give the maximum and minimum powers of A that occur in the Kauffman bracket of the link diagram D* in terms of the corresponding ones of D.

Proposition 3.6. Let D* be the adequate diagram obtained by replacing a crossing of positive sign in the adequate diagram D by a block of rational tangles that extends it consisting of a product of l tangles each of which is a sum of kn rational tangles for 1≤n≤l with corresponding continued fraction [a1ij,a2ij,…,amijij] for 1≤i≤l and 1≤j≤ki, then we have

and

where T+ij=∑r=1sa2r-1ij-12((-1)mij-1+1) and T-ij=∑r=1sa2rij-12((-1)mij+1) with 2s-1≤mij and 2s≤mij. In the case that the block consists of a sum of l tangles each of which consisting of a product of kn rational tangles 1≤n≤l, with corresponding continued fraction [a1ij,a2ij,…,amijij] for 1≤i≤l and 1≤j≤ki, then we just interchange the terms 2∑n=1l(kn-1)-1 and -2l+3 after changing their signs.

Proof. It is not to hard to prove the following facts sAD*=sAD+∑i=1l∑j=1klT+ij+∑n=1l(kn-1), and sBD*=sBD+∑i=1l∑j=1klT-ij+(l-1) in the first case and sAD*=sAD+∑i=1l∑j=1klT+ij+(l-1), and sBD*=sBD+∑i=1l∑j=1klT-ij+∑n=1l(kn-1) in the second case. Now the result follows using the formulas for the maximum and minimum powers of A that occur in the Kauffman bracket of any adequate link diagram D given in [7, Proposition 1].

Remark 3.7. A similar result can be obtained when the crossing is of negative sign if we work with the mirror image of the link diagram. Note that the extreme powers of the Kauffman bracket of the link diagram in this case are the opposite of the extreme powers of the Kauffman bracket of the mirror image of the link diagram and vice versa.

Example 3.8. It is easy to see that the pretzel link diagram D( −2,−2,…,−2︸r−times; 2,2,…,2︸s−times) with r,s≥2 is adequate (see the only example in [7, Page 529]). Here 2, and -2 denote the number of positive and negative crossings, respectively. Therefore, the Montesinos link diagram M((α1,β1),(α2,β2),…,(αr,βr) (γ1,δ1),(γ2,δ2),…,(γs,δs)) with r,s≥2 is adequate by Theorem 3.3. In this notation, αi,βi are coprime positive integers with αi>1 for 1≤i≤r and γi,δi are coprime integers with γiδi<0 and |γi|>1 for 1≤i≤s. This confirms the result of [7, Section 4].

Theorem 3.9. Let L be a homogeneous link with a homogeneous diagram D. If L* is the link obtained from D by replacing any oriented crossing c by a block of oriented rational tangles that extends it, then L* is homogeneous with homogeneous diagram D*.

Proof. Let c be any oriented crossing of the homogeneous diagram D. By taking the mirror image if necessary, we may assume that c corresponds to a positive edge in the Seifert graph according to the scheme in Figure 2 and later use the fact that the diagram is homogeneous iff its mirror image is homogeneous to obtain the required result.

We prove the result first when such an oriented crossing is replaced by an oriented integer tangle that extends it. Let Dm be the link diagram obtained from D after replacing the oriented crossing c by an oriented integer tangle of m vertical or horizontal crossings that extends it. We show that Dm is homogeneous if D is homogeneous. According to orientation, the Seifert graph Gm corresponding to Dm is exactly the Seifert graph G corresponding to D where the edge corresponding to c is replaced by m parallel edges or m collinear edges of the same sign. Therefore, Gm is homogeneous if G is homogeneous.

Now we prove the case when such an oriented crossing is replaced by an oriented rational tangle that extends it. For this end, we let [a1,a2,…,an] to be the continued fraction of some oriented rational tangle that extends c. We use induction on the number of denominators of the continued fraction to prove our claim. The above argument implies the result for n=1. Now the oriented rational tangle of corresponding continued fraction [a1,a2,…,an-1+1] is isotopic to the oriented rational tangle of corresponding continued fraction [a1,a2,…,an-1,1] from the third part of Lemma 2.1 and from the fact that isotoping the last crossing of the first tangle from the (n-1)-th box to the n-th box in the second tangle rotates such a crossing counterclockwise by 90∘ which changes the sign and preserves the type of such crossing since the orientation is preserved. This implies that the diagram obtained by replacing the oriented crossing c by the oriented rational tangle of continued fraction [a1,a2,…,an-1+1] is homogeneous iff the diagram obtained by replacing the oriented crossing c by the rational tangle of continued fraction [a1,a2,…,an-1,1] is homogeneous. Therefore, the result follows using the induction hypothesis and the case when n=1. Finally, the result when the oriented crossing c is replaced by a block of oriented rational tangles is obtained by first replacing the crossing c by an oriented integer tangle that extends c and then replacing each crossing of such oriented tangle by an oriented rational tangle that extends the corresponding crossing.

Corollary 3.10. Let L be a link with a diagram D and let D* be the link diagram obtained from D by replacing any oriented crossing c of D by a block of oriented rational tangles that extends it, then the diagram D is homogeneous iff D* is homogeneous.

Proof. The assumption implies that the orientation on D* induces an orientation on D and vice versa. In other words, the orientation of the crossing c extends the orientation of the given rational tangle. Now the result follows directly since the Seifert graph of D is homogeneous iff the Seifert graph of D* is homogeneous.

Corollary 3.11. Let L be a homogeneous link of a homogeneous diagram D such that det(L0) and det(L1) are not both zero, where L0 and L1 are obtained by smoothing D at some crossing c, then L yields an infinite family of distinct homogeneous links.

Proof. It is not too hard to see that the oriented rational tangle [2n+1] that consists of only one denominator with n∈N extends the oriented crossing c. This implies that set of links {L[2n+1]|n∈N} obtained from L by replacing the oriented crossing c by the oriented rational tangle [2n+1] consists of homogeneous links according to the result of Theorem 3.9. Now according to the assumption and to the result of Lemma 2.2, this set consists of infinitely many links since the elements of this set have infinitely many values of the determinant. Therefore, the set of links {L*} is infinite as it contains the infinite subset of links {L[2n+1]|n∈N}.

As a direct consequence of the fact that the class of positive links is a subclass of the class of homogeneous links, we obtain the following:

Corollary 3.12. Let L be a positive link with a positive link diagram D. If L* is the link obtained from D by replacing any oriented crossing c by block of oriented rational tangles that extends it, then L* is positive link with positive link diagram D*.

Theorem 3.13. Let L be an alternative link with a alternative diagram D. If link L* is the link obtained from D by replacing any oriented crossing c by a block of oriented rational tangles that extends it, then L* is alternative with alternative diagram D*.

Proof. Let c be any oriented crossing of the alternative diagram D. By taking the mirror image if necessary, we can assume that c corresponds to a positive edge in the enhanced checkerboard graph according to the scheme in Figure 3 and later use the fact that the diagram is alternative iff its mirror image is alternative to obtain the required result.

We prove the result first when the oriented crossing is replaced by an oriented oriented integer tangle that extends it. Let Dm be the link diagram obtained from D after replacing the oriented crossing c by an oriented integer tangle of m vertical or horizontal crossings that extends it. We show that Dm is alternative if D is alternative. According to orientation, the enhanced signed graph Φ(Gm) corresponding to Dm is exactly the enhanced signed graph Φ(G) corresponding to D where the edge corresponds to c is replaced by m parallel edges or m collinear edges of the same sign. Therefore, Φ(Gm) is alternative if Φ(G) is alternative.

Now we prove the case when such an oriented crossing is replaced by an oriented rational tangle that extends it. For this end, we let [a1,a2,…,an] to be the continued fraction of some oriented rational tangle that extends c. We use induction on the number of denominators of the continued fraction to prove our claim. The above argument implies the result for n=1. Now the oriented rational tangle of corresponding continued fraction [a1,a2,…,an-1+1] is isotopic to the oriented rational tangle of corresponding continued fraction [a1,a2,…,an-1,1] from the third part of Lemma 2.1 and from the fact that isotoping the last crossing of the first tangle from the (n-1)-th box to the n-th box in the second tangle rotates such a crossing counterclockwise by 90∘ which changes the sign and preserves the type of such crossing since the orientation is preserved. This implies that the diagram obtained by replacing the oriented crossing c by the oriented rational tangle of continued fraction [a1,a2,…,an-1+1] is alternative iff the diagram obtained by replacing the oriented crossing c by the oriented rational tangle of continued fraction [a1,a2,…,an-1,1] is alternative. Therefore, the result follows using the induction hypothesis and the case when n=1. Finally, the result when the oriented crossing c is replaced by a block of oriented rational tangles is obtained by first replacing the crossing c by an oriented integer tangle that extends c and then replacing each crossing of such oriented tangle by an oriented rational tangle that extends the corresponding crossing.

Corollary 3.14. Let L be a link with a diagram D and let D* be the link diagram obtained from D by replacing any oriented crossing c of D by a block of oriented rational tangles that extends it, then the diagram D is alternative iff D* is alternative.

Proof. The assumption implies that the orientation on D* induces an orientation on D and vice versa. In other words, the orientation of the crossing c extends the orientation of the given rational tangle. Now the result follows directly since the enhanced checkerboard graph of D is alternative iff the enhanced checkerboard graph of D* is alternative.

Corollary 3.15. Let L be an alternative link of an alternative diagram D such that det(L0) and det(L1) are not both zero, where L0 and L1 are obtained by smoothing D at some crossing c, then L yields an infinite family of distinct alternative links.

Proof. It is not too hard to see that the oriented rational tangle [2n+1] that consists of only one denominator with n∈N extends the oriented crossing c. This implies that set of links {L[2n+1]|n∈N} obtained from L by replacing the oriented crossing c by the oriented rational tangle [2n+1] consists of alternative links according to the result of Theorem 3.9. Now according to the assumption and to the result of Lemma 2.2, this set consists of infinitely many links since the elements of this set have infinitely many values of the determinant. Therefore, the set of links {L*} is infinite as it contains the infinite subset of links {L[2n+1]|n∈N}.

Example 3.16. The first non-alternating adequate knot in Rolfsen's knot table of 10 crossings or less is the knot 10152. It is also homogeneous and alternative with an adequate, a homogeneous and an alternative diagram given in Figure 6. In this figure, we give such diagram with a marked crossing. Moreover, we use the table in [3] to list some adequate knot diagrams that are obtained from this diagram by replacing the marked crossing by the marked rational tangle of three crossings which shows that the knots 12n679, and 12n680 are adequate. The same idea can be applied to other crossings of such diagram to show that the knots 12n558, 12n688, and 12n689 are also adequate.

Figure 6. Diagram of the knot 10152 with a marked crossing and the diagrams of the knots 12n679, and 12n680, respectively.

On the other hand and after choosing appropriate orientations on the given diagrams, it is not too hard to see that these marked oriented rational tangles in the case of the knots 12n679 and 12n680 extend the marked oriented crossing. Thus and according to Theorem 3.9 and Theorem 3.13, we obtain that these knots are homogeneous, and alternative.

The authors would like to express our appreciation to the editor and the anonymous referees for their in-depth comments, suggestions, and corrections, which have greatly improved the manuscript.