Testing Bi-orderability of Knot Groups

Abstract

Abstract. We investigate the bi-orderability of two-bridge knot groups and the groups of knots with or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with or fewer crossings (of which there are ), previous theorems were only able to determine bi-orderability of of the corresponding knot groups. With our methods we are able to deal with more.

Generalized Torsion in Knot Groups

“…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.…”

Generalized Torsion in Knot Groups

“…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.…”

Residual torsion-free nilpotence, biorderability and pretzel knots

“…A group is said to be biorderable if there exists a total order of the group's elements, invariant under both left and right multiplication. Chiswell, Glass and Wilson proved the following fact, using work of Linnell, Rhemtulla and Rolfsen [25], and it has been instrumental in determining the biorderability of several knot groups; see Clay, Desmarais and Naylor [8], Johnson [20] and Perron and Rolfsen [35].…”

Generalized torsion and decomposition of 3-manifolds

“…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.…”

Generalized Torsion in Knot Groups

“…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.…”

Residual torsion-free nilpotence, biorderability and pretzel knots

“…A group is said to be biorderable if there exists a total order of the group's elements, invariant under both left and right multiplication. Chiswell, Glass and Wilson proved the following fact, using work of Linnell, Rhemtulla and Rolfsen [25], and it has been instrumental in determining the biorderability of several knot groups; see Clay, Desmarais and Naylor [8], Johnson [20] and Perron and Rolfsen [35].…”

Generalized torsion and decomposition of 3-manifolds

“…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.…”

Generalized Torsion in Knot Groups

“…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.…”

Residual torsion-free nilpotence, biorderability and pretzel knots

“…A group is said to be biorderable if there exists a total order of the group's elements, invariant under both left and right multiplication. Chiswell, Glass and Wilson proved the following fact, using work of Linnell, Rhemtulla and Rolfsen [25], and it has been instrumental in determining the biorderability of several knot groups; see Clay, Desmarais and Naylor [8], Johnson [20] and Perron and Rolfsen [35].…”

Generalized torsion and decomposition of 3-manifolds

“…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.…”