Torsion Of Manifolds Research Articles

Differential Topology of Semimetals

The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the torsion of manifolds. Dually, a topological semimetal can be represented by Euler chains from which its surface Fermi arc connectivity can be deduced. These dual pictures, and the link to topological invariants of insulators, are organised using geometric exact sequences. We go beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose local charges are subtle Atiyah-Dupont-Thomas invariants globally constrained by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi arcs.

Homology Group on the Dynamical Trefoil Knot

In this article, we introduce the homology group of the dynamical trefoil knot. Also the homology group of the limit dynamical trefoil knot will be achieved. The knot group of the limit dynamical sheeted trefoil knot is presented. The dynamical trefoil knots of variation curvature and torsion of manifolds on their homology groups are deduced. Theorems governing these relations are obtained.

Reidemeister torsion and group invariants of three-dimensional manifolds

Two formulas are presented in this note. The first is purely algebraic and expresses the first elementary ideal of a finitely generated group in terms of the module of elementary derivatives. The second formula expresses the module of elementary derivatives of the fundamental group of a connected compact three-dimensionalpl -manifold with zero Euler characteristic in terms of the Reidemeister torsion of this manifold.