A clasp-pass move (CP-move) is a local change for knots as shown in Fig. 1. In this paper, we introduce two kinds of clasp-pass moves called self, nonself-clasp-pass moves (SCP, NCP-moves) related to self, nonself-crossings. Moreover, we introduce another two kinds of clasp-pass moves called clasp-pass moves of types II, X (CPII, CPX-moves) related to clasp disks.

Figure 1. Clasp-pass move.

Here, we consider CP-moves for oriented knots. There exist positive and negative clasps as shown in Fig. 2. It is sufficient to consider CP-moves of types (+,+), (+,-), (-,-) as shown in Fig. 3. For convenience, we only consider the case (+,+) throughout this paper. We name the crossings c1, c2, c1′, c2′ as shown in Fig. 3. Then we have the following proposition.

Figure 2. Positive and negative clasps.

Figure 3. Clasp-pass moves for oriented knots.

Proposition 1.1. The crossings c1 and c2 are self-crossings (resp. nonself-crossings) after smoothing either a crossing c1′ or c2′ if and only if the crossings c1′ and c2′ are self-crossings (resp. nonself-crossings) after smoothing either a crossing c1 or c2.

Proof. We can check all the cases (A), (B), (C) to connect the ends of the tangle in a CP-move as shown in Fig. 4. Moreover, we see that the cases (A) and (C) correspond to the case of nonself-crossings and the case (B) corresponds to the case of self-crossings.

Figure 4. Cases (A), (B), (C).

By Proposition 1.1, we can define two kinds of CP-moves as follows. If the crossings c1 and c2 are self-crossings after smoothing either a crossing c1′ or c2′, then we call the CP-move a self-clasp-pass move (SCP-move). If the crossings c1 and c2 are nonself-crossings after smoothing either a crossing c1′ or c2′, then we call the CP-move a nonself-clasp-pass move (NCP-move).

Now, we consider another two kinds of CP-moves related to clasp disks. It is known that every knot bounds a clasp disk which consists of a standard disk and some clasp bands. We consider a clasp disk of a knot with the number of clasp bands greater than or equal to two. Choose two clasp bands from the clasp disk and forget the other clasp bands. We call the two clasp bands type II if both ends of one of the two clasp bands are adjacent. Otherwise, we call two clasp bands type X. See a clasp disk with three clasp bands b1, b2, b3 as shown in Fig. 5. We see that the clasp bands b1, b2 are of type X, b1, b3 are of type X, and b2, b3 are of type II. Moreover, we call CP-moves associated with clasp bands of types II, X, CP-moves of types II, X (CPII, CPX-moves). The following is our main theorem of this paper.

Figure 5. A clasp disk with three clasp bands b1, b2, b3.

Theorem 1.2. Knots K and K' are related by a single SCP-move (resp. NCP-move) if and only if K and K' are related by a single CPII-move (resp. CPX-move).

The HOMFLYPT polynomial P(L y,z)∈Z[y±1,z±1] and the Kauffman polynomial F(L a,b)∈Z[a±1,b±1] of an oriented link L are computed by the following recursive formulas [1, 5, 8]:

For the unknot U, we have P(U)=F(U)=1.

For a triple (L+,L-,L0) of oriented links which are identical except near one point as shown in Fig. 6, we have

Figure 6. Skein triple.

In particular, P(L -1,-1z) is the Alexander-Conway polynomial.

For a quadruple (D+,D-,D0,D∞) of oriented link diagrams which are identical except near one point as shown in Fig. 7, we have

Figure 7. Skein quadruple.

where 2ν=w(D0)-w(D∞) and w(D0), w(D∞) are the writhes of D0, D∞, respectively. In particular, we call (L+,L-,L0), (D+,D-,D0,D∞) a skein triple, a skein quadruple, respectively.

The HOMFLYPT and Kauffman polynomials of an oriented r-component link L are presented by the following:

where pn(L y)∈Z[y±1] and fn(L a)∈Z[a±1]. As mentioned in the paper [7], we have

In particular, p0(L y) is a Laurent polynomial in the variable -y2. Therefore, putting -y2=x, we call it the Γ-polynomial Γ(L x)∈Z[x±1], that is, Γ(L -y2)=p0(L y)=f0(L y). The Γ-polynomial is computed by the following recursive formula [6]:

For the unknot U, we have Γ(U)=1.

For a skein triple (L+,L-,L0), we have

where δ=(r+-r0+1)/2 (=0,1) for the numbers r+, r0 of components of L+, L0, respectively. Moreover, the Γ-polynomial has the following property.

Proposition 2.1. Let L be an oriented r-component link with the components K1,…,Kr and lk(L) the total linking number of L. Then we have

Therefore, we obtain a special skein relation of the Γ-polynomial as follows. For a skein triple (K+,K-,K0=K1∪K2) with oriented knots K+, K-, K1, K2, we have

It is known that the Γ-polynomial is characterized as follows.

Theorem 2.2. ([6]) Let K be the set of oriented knots. The image of K under Γ is the following:

We can prove the following proposition by using the skein relation (1) and Theorem 2.2 easily. It is also shown in the paper [4].

Proposition 2.3. Knots K and K' are related by a single SCP-move, then Γ(K)≠Γ(K′). Knots K and K' are related by a single NCP-move, then Γ(K)=Γ(K′).

Question 2.4. If Γ(K)=Γ(K′), then K and K' are related by a finite sequence of NCP-moves?

For the knot 814 in Rolfsen's table [9] and the unknot U, we have

Moreover, it is known that 814 has a presentation shown in Fig. 8 [10]. We see that 814 is deformed into U by a single NCP-move. The following theorem is famous.

Figure 8. 814 is deformed into U by a single NCP-move.

Theorem 2.5. ([2, 3]) Knots K and K' are related by a finite sequence of CP-moves if and only if a2(K)=a2(K′), where a2(K) is the second coeffient of the Alexander-Conway polynomial.

For the second derivative Γ(2)(K 1) of the Γ-polynomial at x=1, we have

Therefore, if Γ(K)=Γ(K′), then a2(K)=a2(K′). If the answer of Question 2.4 is yes, then we can obtain the Γ-polynomial version of Habiro's theorem.

(i) If K and K' are related by a single CPII-move, then K and K' are related by a single SCP-move.

We consider a clasp disk with two clasp bands of type II as shown in Fig. 9.

Figure 9. CPII ⇒ SCP and CPX ⇒ NCP.

We see that the crossings are self-crossings after smoothing the crossing shown in Fig. 9.

(ii) If K and K' are related by a single CPX-move, then K and K' are related by a single NCP-move.

In a similar way, we consider a clasp disk with two clasp bands of type X as shown in Fig. 9. We see that the crossings are nonself-crossings after smoothing the crossing shown in Fig. 9.

(iii) If K and K' are related by a single SCP-move, then K and K' are related by a single CPII-move.

We consider a knot K which has a presentation shown in Fig. 4 (B). We apply a crossing change and attaching a clasp band at a suitable crossing of K repeatedly as shown in Fig. 10. Then we obtain a knot K^ which has a presentation shown in Fig. 12 (B) with some clasp bands. Here, we put a clasp disk on the knot K^. Then we obtain a singular disk with ribbon and clasp singularities. Moreover, we can change one ribbon singularity to two clasp singularities as shown in Fig 11. Finally, we obtain a desired clasp disk as shown in Fig. 12 (B). We see that the two clasp bands are of type II.

Figure 10. A crossing change and attaching a clasp band.

Figure 11. One ribbon singularity is changed to two clasp singularities.

Figure 12. SCP ⇒ CPII and NCP ⇒ CPX.

(iv) If K and K' are related by a single NCP-move, then K and K' are related by a single CPX-move.

In a similar way, we obtain desired clasp disks as shown in Fig. 12 (A) (C). We see that the two clasp bands are of type X.