arXiv research

Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the sequence of torsion numbers satisfies a linear homogeneous recurrence relation …

Study shows torsion order bounds band-unlinking number for knot cobordisms.

Topological complexity of hyperbolic groups equals twice their torsion number.

Knots without 2-torsion have minimal Khovanov homology rank.

Paper finds generalized torsions in non-bi-orderable 3-manifold groups.

Reidemeister torsion is algebraic for most 3-manifolds.

Study computability of real numbers from group properties.

Defines a new Upsilon torsion function for knot Floer homology.

Proves conjecture about integer sums of torus knot torsions.

Analytic torsion and Reidemeister torsion help show exponential growth of torsion in cohomology.

Groups with specific properties have vanishing $\ell^2$-Betti numbers.

We show that a random 3-manifold with positive first Betti number admits a tower of cyclic covers with exponential torsion growth.

Floer homology bounds knot properties like bridge index and ribbon distance.

In the Khovanov homology of links, presence of $\mathbb{Z}_2$-torsion is a very common phenomenon. Finite number of examples of knots with $\mathbb{Z}_n$-torsion for $n>2$ were also known, none for $n>8$. In this paper, we prove that there are infinite families of links whose Khovanov homology contains $\mathbb{Z}_n$-t…

Simplicial versions of topological abelian gauge theories are constructed which reproduce the continuum expressions for the partition function and Wilson expectation value of linked loops, expressible in terms of R-torsion and linking numbers respectively. The new feature which makes this possible is the introduction o…

For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative tor…

Bounds on homology of hyperbolic orbifolds using simplicial models.

A geometric interpretation of curvature and torsion of linear transports along paths is presented. A number of (Bianchi type) identities satisfied by these quantities are derived. The obtained results contain as special cases the corresponding classical ones concerning curvature and torsion of linear connections.

Following the approach of Bryant we study the intrinsic torsion of a SU(3)-manifold deriving a number of formulae for the Ricci and the scalar curvature in terms of torsion forms. As a consequence we prove that in some special cases the Einstein condition forces the vanishing of the intrinsic torsion.

Homotopy braid groups are shown to be torsion-free.

We obtain a number of results regarding freeness, quasiconvexity and separability for subgroups of Coxeter groups, Artin groups and one-relator groups with torsion.

Introduces a massive variant of Ray-Singer Torsion to avoid zero modes in topological field theories.

We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta functions have as special value the quotient of the holomorphic torsion of Ray an…

Method constructs fundamental domains for Picard modular groups.

We study the intrinsic torsion of almost quaternion-Hermitian manifolds via the exterior algebra. In particular, we show how it is determined by particular three-forms formed from simple combinations of the exterior derivatives of the local Kaehler forms. This gives a practical method to compute the intrinsic torsion a…

We study the Whitehead torsions of inertial h-cobordisms, and identify various types representing a nested sequence of subsets of the Whitehead group. A number of examples are given to show that these subsets are all different in general.

The paper examines 2-torsion in instanton Floer homology for knots and 3-manifolds.

Study 2-complexes' homology properties and torsion growth.

New knot theory module shows torsion-ness in number theory.

The paper supports a conjecture about a vanishing identity for certain 3-manifolds.

Paper introduces simplified formulas for Milnor's triple linking number.

The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vo…

Study tight contact structures on specific 3-manifolds.

The analytic torsion is computed on fixed-point free and non fixed-point free factors (tessellations) of the three--sphere. We repeat the standard computation on spherical space forms (Clifford-Klein spaces) by an improved technique. The transformation to a simpler form of the spectral expression of the torsion on sphe…

The paper analyzes the asymptotic behavior of a knot polynomial for a specific real number.

Tangle replacements help in understanding knot properties.

Analyzes a finite set of metrics and functions to determine manifold torsion.

The paper classifies 3-manifold groups with specific torsion elements.

On an open manifold, the spaces of metrics or connections of bounded geometry, respectively, split into an uncountable number of components. We show that for a pair of metrics or connections, belonging to the same component, relative $ζ$-functions, determinants, torsion for pairs of generalized Dirac operators are well…

Given an $L^2$-acyclic connected finite $CW$-complex, we define its universal $L^2$-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group $\operatorname{Wh}^w(G)$. We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré d…

Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.

Grid homology shows knot unknotting lower bound.

We prove equality between the renormalized Ray-Singer analytic torsion and the intersection R-torsion on a Witt-manifold with cusps, up to an error term determined explicitly by the Betti numbers of the cross section of the cusp and the intersection R-torsion of a model cone. In the first step of the proof we compute e…

Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, the…

Let Mod_{g,b} denote the mapping class group of a surface of genus g with b punctures. Feng Luo asked in a recent preprint if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements suffice to ge…

Study torsion-free nilpotent fundamental groups of smooth varieties up to rank 7.

In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and …

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary. Consider any element φin the first cohomology of M with integral coefficients. Then one can define the φ-twisted L^2-torsion function of the universal covering which is a function from the set of positive real nu…