arXiv subjects

Bao Vuong

Publications and source records attributed to Bao Vuong.

4 recordsLinked to original sources

arXiv2023

Fundamental group and twisted Alexander polynomial of link complement in 3-torus

We consider a diagrammatic approach to investigate tame knots and links in three dimensional torus $T^3$. We obtain a finite set of generalised Reidemeister moves for equivalent links up to ambient isotopy. We give a presentation for fundamental group of link complement in 3-torus $T^3$ and the first homology group. We also compute Alexander polynomial and twisted Alexander polynomials of this class of links.

math.AT↗

arXiv2021

The volume of a spherical antiprism

We consider a spherical antiprism. It is a convex polyhedron with $2n$ vertices in the spherical space $\mathbb{S}^3$. This polyhedron has a group of symmetries $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e. rotation to the angle $π/n$ followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedron in $\mathbb{S}^3$. Then we find relations between its dihedral angles and edge lengths in the form of cosine rules through a property of a spherical isosceles trapezoid. Finally, we obtain an explicit integral formula for the volume of a spherical antiprism in terms of the edge lengths.

math.MG↗

arXiv2021

Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space $\mathbb{H}^3$. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in $\mathbb{H}^3$. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza's formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

math.MG↗

arXiv2018

The volume of a compact hyperbolic antiprism

We consider a compact hyperbolic antiprism. It is a convex polyhedron with $2n$ vertices in the hyperbolic space $\mathbb{H}^3$. This polyhedron has a symmetry group $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e. rotation to the angle $π/n$ followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedra in $\mathbb{H}^3$. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formulas expressing the volume of a hyperbolic antiprism in terms of the edge lengths.

math.MG↗