Testing Bi-orderability of Knot Groups

Let G = x, t | w be a one-relator group, where w is a word in x, t. If w is a product of conjugates of x then, associated with w, there is a polynomial A w (X) over the integers, which in the case when G is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on w, that if all roots of A w (X) are real and positive then G is bi-orderable, and that if G is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the results relies on an extension of work of G. Baumslag on adjunction of roots to groups, and this may have independent interest.

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“…Further applications of Corollary 2.5 to knot groups can be found in [3]. Write w as in ( * ) in the Introduction.…”

Section: Examples and Applicationsmentioning

confidence: 99%

Residual nilpotence and ordering in one-relator groups and knot groups

Chiswell¹,

Glass

Wilson³

2015

Math. Proc. Camb. Phil. Soc.

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“…The last prime knot with at most six crossings is 6 3 : Its Alexander polynomial is 1 − 3t + 5t 2 − 3t 3 + t 4 which has no real roots. Using this fact, it is shown in [3], again using results of [2], that the group of this knot is not bi-orderable. We do not know if its group contains generalized torsion.…”

Section: The First Few Prime Knotsmentioning

confidence: 87%

“…The knot in the tables which follows 5 2 is sometimes known as the "stevedore knot" ... 6 1 : This knot has Alexander polynomial 2−5t+2t 2 , whose roots are 1/2 and 2. According to [3], using results of [2], the group of this knot is bi-orderable. Therefore its group does NOT have generalized torsion.…”

Section: The First Few Prime Knotsmentioning

confidence: 99%

Generalized Torsion in Knot Groups

Naylor¹,

Rolfsen²

2016

Can. math. bull.

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Abstract. In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot 5 2 and algebraic knots in the sense of Milnor.

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“…Proof. Since p i > 0, [6] shows that π 1 (E(K pi )) is bi-orderable, and thus it has no generalized torsion element. Recall that X(m) is a composing space which is a locally trivial circle bundle over a punctured disk, referring [3, Theorem 1.5] we see that X(m) is also bi-orderable, and hence its fundamental group does not have a generalized torsion element.…”

Section: Construction Of Toroidal 3-manifolds With Only Global Genera...mentioning

confidence: 97%

Generalized torsion and decomposition of 3-manifolds

Ito¹,

Motegi²,

Teragaito³

2018

Preprint

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A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable 3-manifold M has a generalized torsion element if and only if the fundamental group of some prime factor of M has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal 3-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements.

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Testing Bi-orderability of Knot Groups

Cited by 19 publications

References 12 publications

Residual nilpotence and ordering in one-relator groups and knot groups

Residual nilpotence and ordering in one-relator groups and knot groups

Generalized Torsion in Knot Groups

Generalized torsion and decomposition of 3-manifolds

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