Infinitely many knots with the same polynomial invariant
Infinitely many knots with the same polynomial invariant
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具有相同多项式不变量的无穷多个结
DOI：
10.1090/s0002-9939-1986-0831406-7
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发表时间：
1986
影响因子：
0.7
通讯作者：
T. Kanenobu
中科院分区：
文献类型：
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作者：
T. Kanenobu
我们给出了S3中无穷多个纽结的例子，其中具有最近发现的相同的二元和Jones多项式，但不同的Alexander模结构，这些结构是双曲的，纤维的，带的，亏格2的带状的，和3-桥的。如果存在S3的保向同胚，则S3中的两个纽结K1和K2属于相同的同构型。我们用K1=K2表示它。1984年，V.Jones[9]发现了定向节或环的等价型的一个非常强大的多项式不变量。随后，由Ocneanu[13]、Lickorish和Millett[12]、Hoste[8]以及Freyd和Yetter同时独立地将Jones多项式推广到二元多项式不变量。在这本笔记中，我们跟随利科里什和米利特。对于一个环L，多项式L(1，m)是由下列两个条件递归定义的：(I)如果L、L和Lo是三个具有完全相同投影的环，除在一个交叉处外，它们如图1所示相关，则LL+(L，m)+1-‘L_(L，m)+Mlo(L，m)=0。(Ii)如果K是平凡纽结，则K(L，m)=1。
We give infinitely many examples of infinitely many knots in S3 with the same recently discovered two-variable and Jones polynomials, but distinct Alexander module structures, which are hyperbolic, fibered, ribbon, of genus 2, and 3-bridge. Two knots K1 and K2 in S3 belong to the same isotopy type if there exists an orientation preserving homeomorphism of S3 which maps K1 onto K2. We denote it by K1 = K2. In 1984, V. Jones [9] discovered a very powerful polynomial invariant of the isotopy type of an oriented knot or link. Subsequently, the Jones polynomial was generalized to the two-variable polynomial invariant simultaneously and independently by Ocneanu [13], Lickorish and Millett [12], Hoste [8], and Freyd and Yetter. In this note we follow Lickorish and Millett. For a link L, the polynomial L (1, m) is defined recursively by the following two conditions: (I) If L?, L_ and Lo are three links with completely identical projections except at one crossing, where they are related as shown in Figure 1, then lL+(l, m) + 1-'L_(l, m) + mLo(l, m) = 0. (II) If K is a trivial knot, then K(l, m) = 1.