The Residual Finiteness of the Classical Knot Groups

Abstract

The purpose of this paper is to extend the class of knot groups whose commutator subgroups are known to be residually a finite pgroup (i.e., residually of order a power of the prime p). Such a knot group is known to be residually finite (see, e.g., [10]), and although this class is quite restricted we will show that it includes all the groups of knots in the classical knot table [15].

3-manifold groups

“…The residual finiteness of fundamental groups of compact 3-manifolds was first shown by Hempel [Hem87] and Thurston [Thu82a, Theorem 3.3]. Some pre-Geometrization results on the residual finiteness of fundamental groups of knot exteriors were obtained by Mayland, Murasugi, and Stebe [May72,May74,May75a,May75b,MMi76,Ste68].…”

Residual Torsion-Free Nilpotence, Bi-Orderability and Pretzel Knots

“…Outline. In section 2, we review Mayland's technique used in [23] to analyze the residual properties of the commutator subgroup of a knot group. In section 3, we apply this technique to genus one pretzel knots and prove Theorem 1.2 and Theorem 1.13.…”

“…In [23], Mayland, used a description of the commutator subgroup of a knot group to study the residual finiteness of knot groups. In this section, we show how Mayland's technique can be used to find a sufficient condition for the commutator subgroup of a knot group to be residually torsion-free nilpotent.…”

“…, k n ) to be the pretzel knot represented in the diagram in Figure 1. In [23], Mayland describes a technique used to examine the commutator subgroup of the knot group of a knot bounding an unknotted minimal genus Seifert surface; see section 2. In fact, this is the technique used by Mayland and Murasugi in [24].…”

“…

The residual torsion-free nilpotence of the commutator subgroup of a knot group has played a key role in studying the bi-orderability of knot groups [27,21,8,18]. A technique developed by Mayland in [23] provides a sufficient condition for the commutator subgroup of a knot group to be residually-torsion-free nilpotent using work of Baumslag [1,2]. In this paper, we apply Mayland's technique to several genus one pretzel knots and a family of pretzel knots with arbitrarily high genus.

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