Non-trivial Torus Research Articles

Orderability of link quandles

AbstractThe paper develops a general theory of orderability of quandles with a focus on link quandles of tame links and gives some general constructions of orderable quandles. We prove that knot quandles of many fibred prime knots are right-orderable, whereas link quandles of most non-trivial torus links are not right-orderable. As a consequence, we deduce that the knot quandle of the trefoil is neither left nor right-orderable. Further, it is proved that link quandles of certain non-trivial positive (or negative) links are not bi-orderable, which includes some alternating knots of prime determinant and alternating Montesinos links. The paper also explores interconnections between orderability of quandles and that of their enveloping groups. The results establish that orderability of link quandles behaves quite differently than that of corresponding link groups.

Graph hypersurfaces with torus action and a conjecture of Aluffi

Generalizing the star graphs of M\"uller-Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the corresponding projective graph hypersurface complement is zero. In contrast, we also show that the Euler characteristic in question can take any integer value for a suitable graph. This disproves a conjecture of Aluffi in a strong sense.

Quantum mechanics of a particle on a torus knot: Curvature and torsion effects

In this paper, we study the subtle effect of constraints on the quantum dynamics of a point particle moving on a non-trivial torus knot. The particle is kept on the knot by the constraints, generated by curvature and torsion. In the Geometry-Induced Potential (GIP) approach, the Schrödinger equation for the system yields new results in particle energy eigenvalues and eigenfunctions, in contrast with existing results that ignored curvature and torsion effects. Our results depend on Γ, parameter that characterizes the global features of both the embedding torus and, more interestingly, the knottedness of the path.

A characterization on separable subgroups of 3‐manifold groups

In this paper, we give a complete characterization on which finitely generated subgroups of finitely generated 3-manifold groups are separable. Our characterization generalizes Liu's spirality character on surface subgroups of closed 3-manifold groups. A consequence of our characterization is that, for any compact, orientable, irreducible and ∂-irreducible 3-manifold M with nontrivial torus decomposition, π 1 ( M ) is locally extended residually finite if and only if for any two adjacent pieces in the torus decomposition of M, at least one of them has a boundary component with genus at least 2.

BEYOND ENDOSCOPY FOR THE RELATIVE TRACE FORMULA II: GLOBAL THEORY

For the group $G=\operatorname{PGL}_{2}$ we perform a comparison between two relative trace formulas: on the one hand, the relative trace formula of Jacquet for the quotient $T\backslash G/T$, where $T$ is a nontrivial torus, and on the other the Kuznetsov trace formula (involving Whittaker periods), applied to nonstandard test functions. This gives a new proof of the celebrated result of Waldspurger on toric periods, and suggests a new way of comparing trace formulas, with some analogies to Langlands’ ‘Beyond Endoscopy’ program.

Coverings of torus knots in S2 × S1 and universals

Let [Formula: see text] be an ordinary fiber of a Seifert fibering of [Formula: see text] with two exceptional fibers of order [Formula: see text]. We show that any Seifert manifold with Euler number zero is a branched covering of [Formula: see text] with branching [Formula: see text] if [Formula: see text]. We compute the Seifert invariants of the Abelian covers of [Formula: see text] branched along a [Formula: see text]. We also show that [Formula: see text], a non-trivial torus knot in [Formula: see text], is not universal.

Dynamics of solutions of the Einstein equations with twisted Gowdy symmetry

Some of the most interesting results on the global dynamics of solutions of the vacuum Einstein equations concern the Gowdy spacetimes whose spatial topology is that of a three-dimensional torus. In this paper certain of these ideas are extended to a wider class of vacuum spacetimes where the spatial topology is that of a non-trivial torus bundle over a circle. Compared to the case of the torus these are topologically twisted. They include inhomogeneous generalizations of the spatially homogeneous vacuum spacetimes of Bianchi types II and V I 0 . Using similar procedures, it is shown that the vacuum solutions of Bianchi type V II 0 are isometric to a class of Gowdy spacetimes, the circular loop spacetimes, thus establishing links between results in the literature which were not previously known to be related to each other.

KNOT GROUPS WITH MANY KILLERS

AbstractThe group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none of which is the automorphic image of another, such that each normally generates the group.

Compact pseudo-Riemannian manifolds with parallel Weyl tensor

It is shown that in every dimension n = 3j + 2, j = 1, 2, 3, . . ., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial torus bundles over the circle. They all arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.

A torus subgroup of the isotropy group of a bounded homogeneous domain

We show that the isotropy group of a bounded homogeneous domain always has a non-trivial torus subgroup, which gives an answer to the problem posed by Hundemer.

Local Heterotic Torsional Models

We present a class of smooth supersymmetric heterotic solutions with a non-compact Eguchi-Hanson space. The non-compact geometry is embedded as the base of a six-dimensional non-Kahler manifold with a non-trivial torus fiber. We solve the non-linear anomaly equation in this background exactly. We also define a new charge that detects the non-Kahlerity of our solutions.

Examples of non-rigid CAT(0) groups from the category of knot groups

C Croke and B Kleiner have constructed an example of a CAT(0) group with more than one visual boundary. J Wilson has proven that this same group has uncountably many distinct boundaries. In this article we prove that the knot group of any connected sum of two non-trivial torus knots also has uncountably many distinct CAT(0) boundaries.

Crosscap numbers of torus knots

The crosscap number of a knot in the 3-sphere is the minimal genus of non-orientable surfaces bounded by the knot. We determine the crosscap numbers of torus knots. Also, we show that a minimal genus non-orientable surface bounded by a non-trivial torus knot is unique.

Cyclic surgery on satellite knots

In [9] L. Moser classified all manifolds obtained by Dehn surgery on torus knots. In particular she proved the following (see also [8, Chapter IV]).Theorem 1 [9]. Nontrivial surgery with slope m/n on a nontrivial torus knot T(p, q) gives a manifold with cyclic fundamental group iff m = npq ± 1 and the manifold obtained is the lens space L(m, nq2).