Polynomials of 2-cable-like links
Polynomials of 2-cable-like links
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2 类电缆链路的多项式
DOI：
10.1090/s0002-9939-1987-0884479-0
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
A. S. Lipson
中科院分区：
文献类型：
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作者：
W. Lickorish;A. S. Lipson
Morton和Short [MS]已经通过实验建立了两个节点K1和K2可以具有相同的二元多项式P（l，m）。（见[FYHLMO]，[LM]）而K1和K2上的2-电缆可以通过P区分。我们在这里证明，如果K1和K2是突变对，然后他们的2-电缆和双打（以及其他在突变缠结边界上的2股卫星）类似的结果是真实的无方向的结多项式Q和它的有方向的两个变量的副本F（见[BLM]，[K]）。如果K1，K2是多个组件的链接，则结果为假。1.你的助手。对于3-球面中的每个定向链接L，存在由以下唯一定义的2变量劳伦多项式PL（1，m）E Z [1 m，mi ']：（i）对于非结U，Pu = 1，（ii）IPL ++ I1PL + mPLo = 0，其中L+，L_和Lo在球外部相同，内部如图1（a）所示（参见[FYHLMO]和[LM]）。类似地，我们可以唯一地为每个无向链接L分配一个1变量的Laurent多项式QL（X）EZ [X '1]，使得：（iii）对于非结U，Qu = 1，（iv）（QL ++ QL X（QLO + QLOO）= 0，其中L+、L、Lo和Loo在球外部和球内部是相同的，如图1（B）所示（参见[BLM]）。假设结K由缠结R、S组成，如图2（a）所示。通过使R绕所示三个轴中的一个旋转而获得的任何结是K的突变体。我们用pR、aR、rR表示R在这些变换下的象，用pK、oK、rK表示K的相应突变体。回想一下，对于任何突变u，PK = Pj，K和QK = QUK。2.结果。定理1.设有向纽结K2是由有向纽结K通过缠结R的突变而获得的。设K ′，K2是关于K1，K2的（2，nr）-索。则K '和K'共享相同的二元多项式P（l，m）。我们证明定理1在？3.在哪？4我们证明了以下几点。定理2.设K1、K2如定理1所示，K '、K2是K1、K2的双索（或（2，2n）-索，其中一个分量的方向已被反转）。则K1和K2共享相同的二元多项式P（l，m）。编辑于1986年3月12日收到。1980年数学学科分类（1985年修订）。第57M25关键字和短语。突变体，缠结，2股卫星，绞链一代。(? 1987年美国数学学会0002 - 9939/87 $1.00 +$.25每页
Morton and Short [MS] have established experimentally that two knots K1 and K2 may have the same 2-variable polynomial P(l, m) (see [FYHLMO], [LM]) while 2-cables on K1 and K2 can be distinguished by P. We prove here that if K1 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 K1, K2 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) E Z[1 m, mi'] defined uniquely by the following: (i) Pu = 1 for the unknot U, (ii) IPL+ + I1PL + mPLo = 0, where L+, L_, and Lo are identical outside a ball and inside are as shown in Figure l(a) (see [FYHLMO] and [LM]). Similarly, we may uniquely assign to each nonoriented link L a 1-variable' Laurent polynomial QL(X) E Z[X'1] such that: (iii) Qu = 1 for the unknot U, (iV) (QL+ + QL X(QLO + QLOO) = 0, where L+, L, Lo, and Loo are identical outside a ball and inside it are as shown in Figure l(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, aR, rR and the corresponding mutants of K by pK, oK, rK respectively. Recall that PK = Pj,K and QK = QUK for any mutation ,u. 2. The results. THEOREM 1. Let the oriented knot K2 be obtained from an oriented knot K, by mutation of the tangle R. Let K', K2 be (2, nr)-cables about K1, K2. Then K' and K' share the same 2-variable polynomial P(l, m). We prove Theorem 1 in ?3. In ?4 we prove the following. THEOREM 2. Let K1, 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 K1, K2. Then K, and K2 share the same 2-variable polynomial P(l, m). Received by the editors March 12, 1986. 1980 Mathematics S*ject Cla&ifJiaion (1985 Revision). Primary 57M25. Key uwds and phrses. Mutant, tangle, 2-stranded satellite, skein generation. (?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page