Polynomials of 2-cable-like Links

Morton and Short [MS] have established experimentally that two knots Kx and K2 may have the same 2-variable polynomial P(l,m) (see [FYHLMO], [LM]) while 2-cables on K\ and Ki can be distinguished by P. We prove here that if K\ and K2 are a mutant pair, then their 2-cables and doubles (and other satellites which are 2-stranded on the boundary of the mutating tangle) cannot be distinguished by P. Similar results are true for the unoriented knot polynomial Q and its oriented two-variable counterpart F (see [BLM], [K]). The results are false if K\, Ki are links of more than one component. 1. Preliminaries. There exists for each oriented link L in the 3-sphere a 2-variable Laurent polynomial Pl(1,m) G Z[l±1,m±1] defined uniquely by the following: (i) Pu = 1 for the unknot U, (ii) IPl+ + I"xPl_ + mPLo = 0, where L+,L-, and Lq are identical outside a ball and inside are as shown in Figure 1(a) (see [FYHLMO) and [LM]). Similarly, we may uniquely assign to each nonoriented link L a 1-variable Laurent polynomial Ql(x) G Zfx*1] such that: (iii) Qu = 1 for the unknot U, (iv) (Ql+ +Ql.)x(QLo +QlJ = 0, where L+,L_,Lq, and Loo are identical outside a ball and inside it are as shown in Figure 1(b) (see [BLM]). Suppose the knot K is made up of tangles R, S, as in Figure 2(a). Any knot obtained by rotating R about one of the three axes shown is a mutant of K. We denote the images of R under these transformations by pR, oR, tR and the corresponding mutants of K by pK, aK, tK respectively. Recall that Pk = PßK and Qk = QpK for any mutation p. 2. The results. THEOREM l. Let the oriented knot K2 be obtained from an oriented knot Kx by mutation of the tangle R. Let K'x,K'2 be (2,n)-cables about Kx,K2. Then K[ and K2 share the same 2-variable polynomial P(l,m). We prove Theorem 1 in §3. In §4 we prove the following. THEOREM 2. Let Kx, K2 be as in Theorem 1 and K[,K2 be doubles (or (2,2n)-cables in which the orientation of one component has been reversed) of Kx,K2. Then K[ and K2 share the same 2-variable polynomial P(l,m). Received by the editors March 12, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25.