Speaker:  Andrei Vesnin (Tomsk State University)

Date and time: 24.02.2021, 17:00 (GMT +03:00)

Title: Hyperbolic polyhedra and hyperbolic knots: the right-angled case

Abstract:  A polyhedron is said to be right-angled if all its dihedral angles are equal to pi/2. Three-dimensional hyperbolic manifolds constructed from bounded right-angled polyhedra have many interesting properties [1]. Inoue [2,3] initiated enumerating bounded right-angled hyperbolic polyhedra by their volumes. Atkinson obtained low and upper bounds of volumes via vertex number [4]. In [5] we enumerate ideal (with all vertices at infinity) right-angled hyperbolic polyhedra. The obtained results imply that the right-angled knot conjecture from [6] holds for knots with small crossing number. Atkinson’s upper bounds were improved in [7,8] for bounded and ideal cases both. Finally, we will discuss the relation of results from [5] with the maximum volume theorem from [9].

The talk is based on joint results with Andrey Egorov [5,7,8].

References.

[1] A. Vesnin, Right-angled polyhedra and hyperbolic 3-manifolds, Russian Mathematical Surveys 72 (2017), 335-374.

[2] T. Inoue, Organizing volumes of right-angled hyperbolic polyhedra, Algebr. Geom. Topol. 8 (2008), 1523-1565.

[3] T. Inoue, Exploring the list the smallest right-angled hyperbolic polyhedra, Experimental Mathematics 2019, published online.

[4] C. Atkinson, Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebr. Geom. Topol. 9 (2009), 1225-1254.

[5] A. Vesnin, A. Egorov, Ideal right-angled polyhedra in Lobachevsky space, Chebyshevskii Sbornik 21 (2020), 65-83.

[6] A. Champanerkar, I. Kofman, J. Purcell, Right-angled polyhedra and alternating links, arXiv:1910.13131.

[7] A. Egorov, A. Vesnin, Volume estimates for right-angled hyperbolic polyhedra, Rendiconti dell’Instituto di Matematica dell’Universita di Trieste 52 (2020), 565-576.

[8] A. Egorov, A. Vesnin, On correlation of hyperbolic volumes of fullerenes with their properties, Comput.  Math. Biophys. 8 (2020), 150-167.

[9] G. Belletti, The maximum volume of hyperbolic polyhedral, Trans. Amer. Math. Soc. 374 (2021), 1125-1153.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com