Related papers: Generalized torsion elements in the knot groups of…

In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a…

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…

We show that a group whose generalized torsion elements are torsion elements (which we call a $TR^{*}$-group) is torsion-by-$R^{*}$ group, an extension of torsion group by a group without generalized torsion elements. We also discuss a…

In a group, a generalized torsion element is a non-identity element whose some non-empty finite product of its conjugates yields the identity. Such an element is an obstruction for a group to be bi-orderable. We show that the Weeks…

We consider the 3-manifold obtained by the 0-surgery along a double twist knot. We construct a candidate for a generalized torsion element in the fundamental group of the surged manifold, and see that there exists the cases where the…

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…

Let $G$ be a group. If an equation $x^n = y^n$ in $G$ implies $x = y$ for any elements $x$ and $y$, then $G$ is called an $R$--group. It is completely understood which knot groups are $R$--groups. Fay and Walls introduced $\bar{R}$--group…

A group element is called a generalized torsion if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the…

A non-trivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for…

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for…

A group element is called generalized torsion if a finite product of its conjugates is equal to the identity. We show that in a finitely generated abelian-by-finite group, an element is generalized torsion if and only if its image in the…

A generalized torsion element is a non-trivial element such that some non-empty finite product of its conjugates is the identity. We construct a generalized torsion element of the fundamental group of a 3-manifold obtained by Dehn surgery…

A generalized torsion in a group, an non-trivial element such that some products of its conjugates is the identity. This is an obstruction for a group being bi-orderable. Though it is known that there is a non bi-orderable group without…

We give a condition sufficient to ensure that an amalgam of groups is generalized torsion-free. As applications, we construct a closed 3-manifold whose fundamental group is generalized torsion-free and non bi-orderable; a one-relator group…

Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the identity, then $g$ is called a generalized torsion element. The minimum number of conjugates in such a product is…

The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not…

Generalized knot groups $G_n(K)$ were introduced independently by Kelly (1991) and Wada (1992). We prove that $G_2(K)$ determines the unoriented knot type and sketch a proof of the same for $G_n(K)$ for $n>2$.

We give a first example of 2-knots with the same knot group but different knot quandles by analyzing the knot quandles of twist spins. As a byproduct of the analysis, we also give a classification of all twist spins with finite knot…

In this paper we discuss generalizations of discrete torsion to noninvertible symmetries in 2d QFTs. One point of this paper is to explain that there are two complementary generalizations. Both generalizations are counted by $H^2(G,U(1))$…

Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted…