A skein relation for the HOMFLYPT polynomials of two-cable links

Algebraic & Geometric Topology 7 (27) 2 232 2 A skein relation for the HOMFLYPT polynomials of two-cable links TAIZO KANENOBU We give a skein relation for the HOMFLYPT polynomials of 2 cable links. We have constructed arbitrarily many 2 bridge knots sharing the same HOMFLYPT, Kauffman, and Links Gould polynomials, and arbitrarily many 2 bridge links sharing the same HOMFLYPT, Kauffman, Links Gould, and 2 variable Alexander polynomials. Using the skein relation, we show their 2 cable links also share the same HOMFLYPT polynomials. 57M25, 57M27 Introduction Soon after the discovery of the HOMFLYPT polynomials (Freyd et al [5] and Przytycki and Traczyk [24]), Morton and Short [22] and Yamada [28] gave examples of a pair of knots with the same HOMFLYPT polynomial that are distinguished by the HOMFLYPT polynomials of their 2 cable links. Moreover, Przytyzki [23], and Lickorish and Lipson [9] showed that if two knots are mutant knots, their 2 cable links share the same HOMFLYPT polynomial; this is false if they are links of more than one component. Recently, Stoimenow [26] found the first examples of four pairs of non-mutant 2 crossing knots whose 2 cable links share the same HOMFLYPT polynomials. They are () f2 34 ; 2 627 g; f2 35 ; 2 872!g; f2 378 ; 2 74 g; f2 423 ; 2 74 g from the table of Hoste, Thistlethwaite, and Weeks [6], where 2 872! is the mirror image of 2 872 and f2 378 ; 2 423 g is a mutant pair. On the other hand, in 992, J R Links and M D Gould [2] discovered a 2 variable polynomial invariant for an oriented link, which we call the LG polynomial. It is known that mutant links share the same LG polynomial (De Wit Links Kauffman [4]) and all prime knots with less than or equal to crossings are completely classified by the LG polynomial (De Wit [2]). By using skein relations given by De Wit et al [4] and Ishii [7], Ishii and the author [8] gave several examples of different knots and links sharing Published: 24 September 27 DOI:.24/agt.27.7.2

22 Taizo Kanenobu the same LG polynomials; the smallest such example is a pair of 2 bridge knots with 4 crossings; see Example 5.4. Then De Wit and Links [3] searched for non-mutant pair of prime knots with and 2 crossings sharing the same LG polynomial, and they discovered such pairs. Surprisingly, they agree with the above knot pairs () found by Stoimenow, that is, the four pairs () are the smallest ones of non-mutant knots whose 2 cable links share the same HOMFLYPT polynomials. This suggests some relation between the HOMFLYPT polynomial of a 2 cable link and the LG polynomial. This motivated the author to discover a skein relation for the HOMFLYPT polynomial of 2 cable links (Theorem 2., Corollary 3.3 and Corollary 3.4), which is similar to one for the LG polynomial. In [8], we have constructed examples of arbitrarily many 2 bridge knots or links sharing the same LG polynomial as well as the HOMFLYPT and Kauffman polynomials [7; 8]; cf Kanenobu [; ; 2; 3; 4] and Kanenobu and Sumi [5; 6]. Our skein relation enables us to prove that their 2 cable links share the same HOMFLYPT polynomials (Theorem 5. and Theorem 5.2). Also, we can construct some other examples with the same 2 cable HOMFLYPT polynomials, which are special case considered in Ishii and Kanenobu [8]. This paper consists of five sections. In Section 2, we prove the skein relation for the 2 cable HOMFLYPT polynomial (Theorem 2.). In Section 3, we define the 2 cable links, give some properties on the HOMFLYPT polynomial (Proposition 3. and Proposition 3.2), and give two corollaries for Theorem 2. (Corollary 3.3 and Corollary 3.4). In Section 4, we consider the 2 cable HOMFLYPT polynomials of the link KŒˇI R ; R 2 ; : : : ; R n, which had been given in [8]. In Section 5, we prove the above-mentioned Theorem 5. and Theorem 5.2. 2 Skein relation The HOMFLYPT polynomial P.LI v; z/ 2 Z Œv ; z (Freyd et al [5], Jones [9] and Przytycki and Traczyk [24]) is an invariant of the isotopy type of an oriented link L, which is defined by the following formulas: (2) (3) v P.L C I v; z/ P.U I v; z/ D I vp.l I v; z/ D zp.l I v; z/: where U is the unknot and L C, L, L are three links that are identical except near one point where they are as in Figure. Equation (3) implies the following, which we

A skein relation for the HOMFLYPT polynomials of two-cable links 23 will use often. (4) (5) P.L C I v; z/ D v 2 P.L I v; z/ C vzp.l I v; z/; P.L I v; z/ D v 2 P.L C I v; z/ v zp.l I v; z/: L C L L Figure Let L.t C /, L.t /, L.e C /, L.e /, L.f C /, L.f /, L.f / be oriented links identical outside a ball and inside are 8 end tangles t C, t, e C, e, f C, f, f as shown in Figure 2, respectively. We denote the HOMFLYPT polynomial of the link L.s/ by P.s/, where s is one of these tangles. Theorem 2. (6) v 5 P.t C / C v 5 P.t / D v 3 P.e C / C v 3 P.e / C v 3 P.f C / C v C v P.f / C v 3 P.f / z 2 : t C t e C e f C f f Figure 2 Remark 2.2 We denote the LG polynomial of an oriented link by LG.L/, which is a 2 variable polynomial in variables t and t. The formula (6) is similar to the skein

24 Taizo Kanenobu relation for the LG polynomial [8, Equation (3)]: LG.L C2 / C t t LG.L 2 / D.t t C /LG.L / 2.t /.t /LG.L /; which is obtained from the relations [8, Equations () and (2)] given by De Wit et al [4] and Ishii [7], respectively. Here, L C2, L 2, L, L are four oriented links that are identical except near one point where they are as in Figure 2. Except for the 8 end tangles given in Figure 2, we use tangles as shown in Figure 3. Note that these tangle diagrams have only positive crossings. e f f 2 f 3 g g 2 g 3 g 4 h h 2 h 3 h 4 h 5 Figure 3 Lemma 2.3 Let t be the 8 end tangle as shown in Figure 4. Then we have: (7) P.t / Dv 8 P.e / C v 7 P.g / C v 5 P.g 2 / C v 5 P.g 3 / C v 3 P.g 4 / z C v 4 P.f C / C v 2 P.f 3 / z 2 :

A skein relation for the HOMFLYPT polynomials of two-cable links 25 c c 4 c 2 c 3 Figure 4 Proof First, we consider the right-hand side of (7). Applying (4) two times, we have (8) P.f 3 / D v 4 P.f / C v 3 zp.f / C v 3 zp.f 2 / C v 2 z 2 P.f C /; Thus the right-hand side of (7) is equal to (9) v 8 P.e / C v 4 z 2 C v 4 z 4 P.f C / C v 6 z 2 P.f / C v 5 z 3 P.f / C v 5 z 3 P.f 2 / C v 7 P.g / C v 5 P.g 2 / C v 5 P.g 3 / C v 3 P.g 4 / z: Next, we consider the left-hand side of (7), P.t /. For ı D.ı ; ı 2 ; ı 3 ; ı 4 /, where each ı i is either a minus sign or zero, we denote by t ı the 8 end tangle obtained from t by changing the positive crossing c i to a negative crossing or smoothing according as ı i is a minus sign or zero. Then applying (4), we have t t t () P.t / D X ı v 2m ı.vz/ n ı P.t ı /; where m ı and n ı are the numbers of minus signs and zeros in ı, respectively. Applying (4) and t (5), we obtain the following: t t t t t () P.t / D P @ A D v 2 P.f / C vzp.f /I (2) P.t t / D P.t / D P.h 3 /I t t t (3) P.t / D P.f /I P.t / D P @ A t t t

t t t t t t 26 Taizo Kanenobu t t t (4) D v 2 P @ t t t v zp @ D v 2 v 2 P.f 2 / C vzp.f C / t t t t t t (5) P.t / D P.h /I A A v zp D v zp.f C / vzp.f / z 2 P.f / C P.f 2 /I (6) P.t / D P @ t t t D v 2 P @ v 2 P.f / C vzp.f / t t t t t t t t D v 2 v 2 P.f / Ct vzp.f / v zp.h 3 / t D P.f / C v t zp.f / v zp.h 3 /I t t t t (7) P.t / D P @ A A v zp.h 3 /: A D v 2 P.h 4 / v zp.h 3 /I t (8) t P.t / D P.h 2 /I t t t t (9) P.t / D P @ t (2) t P.t t / D P @ t t t t t t (2) P.t / D P.g /I (22) P.t / D P @ (23) P.t / D P @ A t D v 2 P.f C / v zp.f /I t t A D v 2 P.h 5 / v zp.h 3 /I t t t A D v 2 P.g 2 / v zp.h /I t t t A t t t t t t t t t t t t

A skein relation for the HOMFLYPT polynomials of two-cable links 27 (24) P.t D v 4 P.g 4 / v 3 zp.h 4 / v 3 zp.h 5 / C v 2 z 2 P.h 3 /I / D P @ A D v 2 P.g 3 / v zp.h 2 /I t (25) t t P.t / D P.e /: Substituting () (25) into (), we obtain (9). This completes the proof. Proof of Theorem 2. By adding two negative crossings to an 8 end tangle as in Figure 5(a), t, e, g, g 2, g 3, g 4, f C, f 3 become t C, e, k, k 2, k 3, k 4, f, f C, respectively, where k ; : : : ; k 4 are tangles as shown in Figure 6. Thus (7) becomes (26) P.t C / Dv 8 P.e / C v 7 P.k / C v 5 P.k 2 / C v 5 P.k 3 / C v 3 P.k 4 / z v 4 P.f / C v 2 P.f C / z 2 : t t t t t C t (a) (b) Figure 5 k k 2 k 3 k 4 k 5 k 6 k 7 k 8 Figure 6

28 Taizo Kanenobu Similarly, adding two negative crossings to an 8 end tangle as in Figure 5(b), we obtain (27) P.t C / Dv 8 P.e / C v 7 P.k 5 / C v 5 P.k 6 / C v 5 P.k 7 / C v 3 P.k 8 / z C v 4 P.f / C v 2 P.f C / z 2 ; where k 5 ; : : : ; k 8 are tangles as shown in Figure 6. Since k 5, k 6, k 7, k 8 are the mirror images of k 4, k 3, k 2, k, respectively, by taking mirror images, (27) becomes (28) P.t / Dv 8 P.e C / C C v 7 P.k 4 / v 5 P.k 3 / v 5 P.k 2 / v 3 P.k / v 4 P.f / C v 2 P.f / In fact, the HOMFLYPT polynomial of the mirror image L! of a link L is obtained from that of L by the formula z 2 : (29) P.L!I v; z/ D P.LI v ; z/: Combining (26) and (28), we obtain (6). z 3 Two-cable links Let L D K [ [ K n be an oriented link with n components and N i be a tubular neighborhood of K i such that N ; : : : ; N n are disjoint. For integers p ; : : : ; p n, let T.p i / be a torus link of type.2; p i / on a solid torus V i. Let ' i W V i! N i be a faithful homeomorphism, that is, the homeomorphism ' i sends the standard meridian-longitude system of V i to a standard meridian-longitude system of N i. Then we call the link '.T.p // [ [ ' n.t.p n // the 2 cable link about L with framing p, p D.p ; : : : ; p n /, which we denote by zl p or zk p [ [ zk p n n ; cf Rolfsen [25, Section 4D]. We assume that the strands in zl p are oriented so that when they are stuck together to make the companion link L, their directions are parallel and agree with the orientation of L We present a 2 cable link zk p [ [ zk p n n by drawing a diagram of the link K [ [ K n together with the framings p ; : : : ; p n near their respective components. We describe this construction diagrammatically. Let n, m be positive integers. In a diagram circles labeled n,, n,, =m, =m stand for an n tangle, a tangle, a n tangle, an tangle, a =m tangle, a =m tangle, respectively, as shown in Figure 7.

... A skein relation for the HOMFLYPT polynomials of two-cable links 29...... n crossings n crossings (a) (b) (c) (d) m crossings... m crossings (e) (f) Figure 7: (a) n tangle, (b) tangle, (c) n tangle, (d) tangle, (e) =m tangle, (f) =m tangle. Figure 8 shows a diagram of the torus link of type.2; p/, T.p/. Let K be an oriented knot with diagram as shown in Figure 9(a), where s is a 2 end tangle. Suppose that the writhe or algebraic crossing number of this diagram is w, that is, the number of positive crossings minus that of negative crossings. Then the 2 cable link about K with framing p has a diagram as in Figure 9(b), where m D 2w p and the strands and crossings in s are transformed in the 4-end tangle diagram zs as shown in Figure 9(c). p Figure 8: Torus link of type.2; p/, T.p/. Let L D K [ K 2 be an oriented 2 component link with diagram as shown in Figure (a), where t is a 4 end tangle and the lower strand belongs to K and the upper one to K 2. Let w i be the writhe of K i in this diagram, i D, 2, that is, the number of positive self-crossings of K i minus that of negative self-crossings of K i. Then the 2 cable link about L with framing.p ; p 2 / has an diagram as in Figure (b), where

22 Taizo Kanenobu s zs ) m ) (a) (b) (c) Figure 9 m i D 2w i p i and the strands and crossings in t are transformed as in the above. Note that the writhe of L is w C w 2 C 2l, where l is the linking number of K and K 2 ; l D lk.k ; K 2 /. m 2 K 2 t zt K m (a) (b) Figure Now we give some properties of the HOMFLYPT polynomials of 2 cable links. Let L D K [ M be a link, where K is a component and M is a complement of K in L. We consider the HOMFLYPT polynomial of its 2 cable link zl p D zk p [ zm, where zm is a 2 cable link of M.

A skein relation for the HOMFLYPT polynomials of two-cable links 22 For an integer k, we define a symmetric polynomial k.v; z/ 2 Z Œv ; z as follows: 8 ' k k if k > ; ˆ< ' (3) k.v; z/ D if k D ; ˆ:. / k v 2k ' k k D. / k v 2k k.v; z/ if k <, ' where ', 2 Z Œv ; z are defined by ' D prove the following by induction. v 2 and ' C D z. Then we can Proposition 3. For an integer k, we have (3) P. zl k / D kp. zl / C v 2 k P. zl /: Thus the HOMFLYPT polynomial of a 2 cable link with framing.k ; : : : ; k n /, k i 2 Z, is obtained from those of the 2 cable links with framings. ; : : : ; n /, i D,. Next we consider the HOMFLYPT polynomials of 2 cable links of a composite link. Let L D J [ M and L 2 D K [ M 2 be oriented links, where J, K are components and M, M 2 are complements of J, K in L, L 2, respectively, possibly empty. We construct a composite link by connecting the components J and K; L #L 2 D J #K [ M [ M 2. We put (32) (33) (34) zl j D zj j [ zm I zl k 2 D zk k [ zm 2 I zl l D.A J #K/ l [ zm [ zm 2 ; where zm, zm 2 are 2 cable links about M, M 2, respectively. Then we have the following proposition. Proposition 3.2 For integers j, k, l with l D j C k, we have (35). 2 /P. zl l / D P. zl j /P. zl k 2 / C P. zl j /P. zl kc 2 / P. zl j /P. zl kc 2 / C P. zl j /P. zl k 2 / ; where D.v link. v/z is the HOMFLYPT polynomial of the trivial 2 component

222 Taizo Kanenobu Proof For the diagrams of L, L 2, L #L 2 given as in Figure (a), where s and t are 2 end tangles, we may give the diagrams of their 2 cable links zl j, zl k 2, zl l as in Figure (b), where zs and zt are 2 cable tangles about s and t, respectively. The link zl l is isotopic to the link given in Figure (c), which is the sum of two 4 end tangles. By Lickorish and Millett [2, Proposition 2] the HOMFLYPT polynomial of the link given in Figure (c) is calculated from the four links zl j, zl j, zl k 2, zl kc 2 as shown in Figure (d), and we obtain (35). s t s t (a) J K J #K L L 2 L #L 2 zs zt zs zt (b) zl j zl k 2 zl l zs ~ zt (c) zs zs zt zt (d) zl j zl j zl k 2 zl kc Figure We apply Theorem 2. to 2 cable links. Let L C2, L 2, L, L be four oriented links that are identical except near one point where they are as in Figure 2. We have two cases: Case : The two strands of L s, s D C2, 2,, in Figure 2 belong to the same component, and those of L belong to different components. Case 2: The two strands of L s, s D C2, 2,, in Figure 2 belong to different components, and those of L belong to the same component.

A skein relation for the HOMFLYPT polynomials of two-cable links 223 L C2 L 2 L L Figure 2 Case We put (36) (37) L s D K s [ M I L D J [ J 2 [ M; where M is the common sublink of L s and L, possibly empty and K s, J, J 2 are the visible components in Figure 2. For integers p, q, we put their 2 cable links as follows. (38) (39) where zl p s D zk p s [ zm I zl.q;r/ D zj q [ zj r 2 [ zm ; zm is a 2 cable link about M. Then from Theorem 2., we have: Corollary 3.3 (4) v 5 P. zl pc2 C2 / C v5 P. zl p 2 2 / D v 3 P. zl pc2 / C v 3 P. zl p 2 / C v 3 P. zl.qc;rc/ / C v C v P. zl.q;r/ / C v 3 P. zl where p D q C r C 4lk.J ; J 2 /..q ;r / / z 2 ; Case 2 We put (4) (42) L s D J s [ K s [ M I L D H [ M; where M is the common sublink of L s and L, possibly empty and J s, K s, H are the visible components in Figure 2. For integers p, q, we put their 2 cable links as follows. (43) (44) zl.p;q/ s D zj s p [ zk s q [ zm I zl r D zh r [ zm ;

224 Taizo Kanenobu where zm is a 2 cable link about M. Then from Theorem 2., we have the following corollary. Corollary 3.4 (45) v 5 P. zl.p ;q / C2 C v 3.p ;q / P. zl / C where r D p C q C 4lk.J ; K /. / C v 5 P. zl.pc;qc/ 2 / D v 3 P. zl.pc;qc/ v 3 P. zl rc2 v / C C v / P. zl r / C v3 P. zl r 2 / z 2 ; 4 The two-cable HOMFLYPT polynomials of the link KŒˇI R ; R 2 ; : : : ; R n In [8, Section 4], for a pure 3 braid ˇ and tangles R, R 2 ; : : : ; R n, R n, we defined a class of oriented links KŒˇI R ; R 2 ; : : : ; R n ; R n as shown in Figure 3; if n D, we interpret it as the 2 bridge knot KŒˇ as shown in Figure 3(c). (a) ˇ R ˇ ˇ Rn ˇ ˇ R 2 R n (b) ˇ R ˇ ˇ ˇ R n ˇ R 2 R n (c) ˇ Figure 3: The link KŒˇI R ; R 2 ; : : : ; R n (c) n D. ; R n with (a) n even, (b) n odd, We say that a 3 braid is strongly amphicheiral if it is of the form (46). q 2 q 2 /. q 2 2 q /I the closure of such a 3 braid is strongly amphicheiral in the sense of Van Buskirk [27]. Here and 2 are elementary 3 braids as shown in Figure 4. Note that if a strongly

A skein relation for the HOMFLYPT polynomials of two-cable links 225 amphicheiral 3 braid ˇ can be oriented as in Figure 3(c), then it is easy to see that ˇ is a pure braid. D D 2 D 2 D Figure 4: Elementary 3 braids. For r i 2 Z [ fg, we denote by KŒˇI r ; r 2 ; : : : ; r n the link KŒˇI R ; R 2 ; : : : ; R n with each tangle R i the r i tangle. If ˇ is a strongly amphicheiral pure 3 braid and each r i is an even integer, then KŒˇI r ; r 2 ; : : : ; r n is a 2 bridge knot or link (with 2 components) according as n is even or odd. Note that a 2 bridge link is interchangeable, that is, there is an isotopy of the 3 sphere which interchange the two components, and so we do not have to distinguish the components. We will prove the following theorem on 2 cable links about the 2 bridge links KŒˇI r ; r 2 ; : : : ; r n. Theorem 4. If ˇ is a strongly amphicheiral pure 3 braid and r, r 2 ; : : : ; r n, r n are even integers, then (47) P. zkœˇi r ; r 2 ; : : : ; r n ; r n p / D P. zkœˇi r n ; r n ; : : : ; r 2 ; r p / for any framing p. In order to prove this theorem, we need the following lemmas: Lemma 4.2 is a special case of [8, Lemma 4.2], and for Lemma 4.3, compare [8, Lemma 4.3]. Lemma 4.2 Suppose ˇ is a pure 3 braid. If r k D, k n, then KŒˇI r ; : : : ; r n is isotopic to: 8 the trivial 2 component link if n D k D ; ˆ< KŒˇI r 3 ; : : : ; r n if n 2, k D ; KŒˇI r ; : : : ; r k 2 ; r k C r kc ; r kc2 ; : : : ; r n if 2 k n ; ˆ: KŒˇI r ; : : : ; r n 2 if n 2, k D n, where both KŒˇI r 3 ; : : : ; r n and KŒˇI r ; : : : ; r n 2 with n D 2 mean KŒˇ. Lemma 4.3 Suppose that ˇ is a strongly amphicheiral pure 3 braid. If r k D, k n, then KŒˇI r ; r 2 ; : : : ; r n is isotopic to the connected sum KŒˇI r ; r 2 ; : : : ; r k #KŒˇI r kc ; r kc2 ; : : : ; r n ;

226 Taizo Kanenobu where both KŒˇI r ; : : : ; r k KŒˇ. with k D and KŒˇI r kc ; : : : ; r n with k D n mean Proof of Theorem 4. We prove by induction on n. The case n D is trivial. Let m be a fixed integer with m 2. Assuming that (47) holds for n < m, we will prove (47) with n D m. Claim If r k D for some k, k m, then (47) with n D m holds for any even integers r j, j k, any strongly amphicheiral pure 3 braid ˇ, and any framing p. Proof This follows from Lemma 4.2 and inductive hypothesis that (47) holds for n < m. Claim 2 If r k D for some k, k m, then (47) with n D m holds for any even integers r j, j k, any strongly amphicheiral pure 3 braid ˇ, and any framing p. Proof There are four cases: Case. m is even and k is even. Case 2. m is even and k is odd. Case 3. m is odd and k is even. Case 4. m is odd and k is odd. We prove only for Case, since other cases are similar. We put K D KŒˇI r ; : : : ; r m, K 2 D KŒˇI r m ; : : : ; r. By Lemma 4.3, K is isotopic to the connected sum of the 2 bridge link KŒˇI r ; : : : ; r k and the 2 bridge knot KŒˇI r kc ; : : : ; r m, which we denote by L and J, respectively, and K 2 is isotopic to the connected sum of the 2 bridge link KŒˇI r k ; : : : ; r and the 2 bridge knot KŒˇI r m ; : : : ; r kc, which we denote by L 2 and J 2, respectively. Thus K s D L s #J s, s D, 2. By Proposition 3.2, we have. 2 /P. zk s k / D P. zl.l;l / s /P. zj s j / C P. zl.l ;l / (48) s /P. zj s jc / P. zl.l;l / s /P. zj s jc / C P. zl.l ;l / s /P. zj s j / ; where k D l Cj. By the inductive hypothesis, P. zl.l;l / / D P. zl.l;l / 2 / for any integers l, l, and P. zj j s / D P. zj j s / for any integer j. Thus we have P. zk k / D P. zk k 2 /, completing the proof.

A skein relation for the HOMFLYPT polynomials of two-cable links 227 Claim 3 Let k m. Suppose that (47) with n D m and r k D r kc D D r m D 2 holds for any even integers r ; : : : ; r k, any strongly amphicheiral pure 3 braid ˇ, and any framing p. Then (47) with n D m and r kc D D r m D 2 holds for any even integers r ; : : : ; r k ; r k, any strongly amphicheiral pure 3 braid ˇ, and any framing p. Proof We prove by induction on r k. There are four cases: Case. m is even and k is even. Case 2. m is even and k is odd. Case 3. m is odd and k is even. Case 4. m is odd and k is odd. We prove only for Case, since other cases are similar. In this case both of (49) (5) KŒˇI r ; : : : ; r k ; r; 2; : : : ; 2 ; ƒ m k KŒˇI 2; : : : ; 2; r; r ƒ k ; : : : ; r m k are 2 bridge knots, so we denote their 2 cable links with framing p by zh p r, zj p r, respectively. We show (5) P. zh p r / D P. zj p r / for any integer p by induction on an even integer r. The case r D follows from Claim. The case r D 2 is the condition of Claim 3. By (4) in Corollary 3.3, we have (52) v 5 P. zh qc2 rc2 / v 3 P. zh r qc2 / v 3 P. zh r q 2 / C v 5 P. zh q 2 D v 3 P. zh.ic;jc/ / C v C v P. zh.i;j/ / C v 3 P. zh (53) v 5 P. zj qc2 rc2 / v 3 P. zj r qc2 / v 3 P. zj r q 2 / C v 5 P. zj q 2 D v 3 P. zj.ic;jc/ / C v C v P. zj.i;j/ / C v 3 P. zj r 2 / r 2 /.i ;j / /.i ;j / / z 2 I z 2 ; where q D i C j C 2.r C r 3 C C r k 3 C r k /. By Claim 2, the right hand side of (52) and that of (53) are equal. Thus if (5) holds for r D l; l C 2, then (5) holds for r D l 2; l C 4. This completes the proof of (5).

228 Taizo Kanenobu Since (47) with n D m and r D r 2 D D r m D 2 is trivial, by induction on k Claim 3 completes the proof of (47) with n D m and any even integers r, r 2 ; : : : ; r m. This completes the proof of Theorem 4.. For integers r i ( ), we denote by KŒˇI =r ; =r 2 the link KŒˇI R ; R 2 with R i the =r i tangle. If each r i is even, then KŒˇI =r ; =r 2 is a knot. We can prove the following in a similar way to Theorem 4.. Theorem 4.4 If ˇ is a strongly amphicheiral pure 3 braid and r, r 2 are even integers, then the 2 cable links with the same framing of the knots (54) KŒˇI =r ; =r 2 ; KŒˇI =r 2 ; =r share the same HOMFLYPT polynomial. Corollary 4.5 The pair of knots.54/ share the same HOMFLYPT, Kauffman, LG polynomials and their 2 cable links share the same HOMFLYPT polynomials. Proof For the HOMFLYPT and Kauffman polynomials, we can prove in a similar way to Kauffman [3, Propositions and 3], respectively. For the LG polynomial, this follows from [8, Theorem 4.]. Example 4.6 According to Corollary 4.5, it is not easy to distinguish the knot pairs (54). For a given pair, we can distinguish by applying the computer program SnapPea of Jeffrey R Weeks. For example, the hyperbolic volumes of (55) KŒ2 2 2 I =2; =2 ; KŒ 2 2 2 I =2; =2 are 8:27794698 and 8:25288455, respectively. Remark 4.7 For a pure 3 braid ˇ and tangles R, R 2 ; : : : ; R n, with n even, let LŒˇI R ; R 2 ; : : : ; R n be an oriented link as shown in Figure 5 [8, Figure ]; for an oriented knot J and an integer p, let ŒˇI R ; R 2 ; : : : ; R n I J; p be the satellite link with companion J, pattern LŒˇI R ; R 2 ; : : : ; R n, and twisting number p as defined in [8, page 282]. If each tangle R i is an r i tangle, where r i is an even integer, we denote these links by LŒˇI r ; r 2 ; : : : ; r n and ŒˇI r ; r 2 ; : : : ; r n I J; p, which are 3-component link. We can prove that the 2 cable links of LŒˇI r ; r 2 ; : : : ; r n and LŒˇI r n ; : : : ; r 2 ; r (resp. ŒˇI r ; r 2 ; : : : ; r n I J; p and ŒˇI r n ; : : : ; r 2 ; r I J; p ) with the same framing share the same HOMFLYPT polynomial in a similar way to [8, Theorem 5.] and Theorem 4..

A skein relation for the HOMFLYPT polynomials of two-cable links 229 R ˇ R ˇ n ˇ ˇ R 2 R n Figure 5: The link LŒˇI r ; r 2 ; : : : ; r n. 5 The two-cable HOMFLYPT polynomials of 2 bridge links In this section, we show the following theorems using Theorem 4.. See Conway [] and Lickorish and Millett [2] for the skein equivalence; if two links are skein equivalent, then they share the same HOMFLYPT polynomial. Theorem 5. For any positive integer N, there exist 2 N, mutually distinct, amphicheiral, fibered 2 bridge knots, which are skein equivalent, share the same Kauffman and LG polynomials, and whose 2 cable links with the same framing share the same HOMFLYPT polynomial. Theorem 5.2 For any positive integer N, there exist 2 N, mutually distinct, amphicheiral, fibered 2 bridge links, which are skein equivalent, share the same Kauffman, 2 variable Alexander, LG polynomials, and whose 2 cable links with the same framing share the same HOMFLYPT polynomial. For a 3 braid and integers p, p 2 ; : : : ; p n (56) (57) Then,, p n, we define 3 braids as follows: hp i D.p ; p / D s p 2 s p I hp ; p 2 ; : : : ; p n ; p n i D hp ; p 2 ; : : : ; p n ihp n i: (58) hp ; p 2 ; : : : ; p n ; p n i D hp ; p 2 ; : : : ; p i ihp i ; : : : ; p n i; where 2 i n, and (59) hp ; p 2 ; : : : ; p n i D.q ; q 2 ; : : : ; q m ; q m /; where m D 3 n and p i D j q j with j.mod 3 i /, j 6.mod 3 i /, and ( if j=3 i.mod 3/; (6) j D if j=3 i 2.mod 3/;

23 Taizo Kanenobu cf [3, page 285]. Using Theorem 4., we can show the following lemma in a similar way to the proof of [8, Lemma 6.3]. Lemma 5.3 Suppose ˇ is a strongly amphicheiral pure 3 braid and b, b 2 ; : : : ; b n are integers. Then for any framing p, we have (6) P. zkœˇh2b ; 2b 2 ; : : : ; 2b n i p / D P. zkœˇh 2b ; 2b 2 ; : : : ; 2b n i p /: For a nontrivial pure strongly amphicheiral 3 braid ˇ, we define the following sets consisting of 2 N 2 bridge knots and links: (62) (63) Kˇ;N D fkœˇh2 ; 2 2 ; : : : ; 2 N i j i D gi Lˇ;N D fkœˇh2 ; 2 2 ; : : : ; 2 N i.2/ j i D g: Proof of Theorem 5. In [8], we have shown that the 2 N knots in Kˇ;N are mutually distinct, amphicheiral, fibered 2 bridge knots, which are skein equivalent, share the same Kauffman and LG polynomials. We then show that their 2 cable links share the same HOMFLYPT polynomial. In fact, for each i, i N, we can show the following in the same way as [8, Equation (4)] using (56) and Lemma 5.3. (64) P. zkœˇh2 ; 2 2 ; : : : ; 2 i ; 2 i ; 2 ic ; : : : ; 2 N i p / This completes the proof. D P. zkœˇh2 ; 2 2 ; : : : ; 2 i ; 2 i ; 2 ic ; : : : ; 2 N i p /: Proof of Theorem 5.2 The 2 bridge links in the above set Lˇ;N are the desired ones. Example 5.4 According to [8, page 286], there are pairs of 2 bridge knots and one pair of 2 bridge links through 2 crossings sharing the same HOMFLYPT, Kauffman, and LG polynomials; for 2 bridge knots, n o KŒ2 2 2 I p; q ; KŒ 2 2 2 I q; p ; n o KŒs 2 2 s2 s 2 2 s 2 I 2; 2 ; KŒs2 2 s2 s 2 2 s 2 I 2; 2 ; where.p; q/ D.2; 2/,.4; 2/,.4; 2/,.6; 2/,.6; 2/,.4; 4/,.8; 2/,.8; 2/,.6; 4/,.6; 4/; and for 2 bridge links n o KŒs 2 2 s 2 I 2; 2; 2 ; KŒs 2 2 s 2 I 2; 2; 2 : By Theorem 4., their 2 cable links also share the same HOMFLYPT polynomials.

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