arXiv 0706.18072007-06math.GT

广义纽结群区分方形结和奶奶结（附大卫·萨维特的附录）

Generalised knot groups distinguish the square and granny knots (with an appendix by David Savitt)

Christopher Tuffley

备注：: 25 pages, 5 figures, to appear in JKTR. v3: example of the target groups added; slight correction to the construction of the target groups; references updated; some changes to notation. v2: section 4.2 expanded to give overview of proof
发表信息：: J. Knot Theory Ramifications 18(8):1129-1157 (2009)
DOI：: 10.1142/S0218216509007385

摘要：

给定一个纽结K，我们可以从K的基本群通过邻接与相应经度交换的子午线的n次方根构造群G_n（K）。这些“广义纽结群“是由Wada和Kelly独立引入的，并且包含基本群作为子群。正方形结SK和奶奶结GK是一对具有同构基本群的不同结的众所周知的例子。我们证明了G_n（SK）和G_n（GK）对所有n>1是非同构的。这证实了Lin和Nelson的一个猜想，并表明G_n（K）（n>1）的同构型比基本群的同构型携带更多关于K的信息.大卫·萨维特的附录包含了证明所需的PSL（2，p）中三叶群表示的一些结果。

英文摘要：

Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalised knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well known example of a pair of distinct knots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) are non-isomorphic for all n>1. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of G_n(K), n>1, carries more information about K than the isomorphism type of the fundamental group. An appendix by David Savitt contains some results on representations of the trefoil group in PSL(2,p) that are needed for the proof.

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