Coxeter groups and meridional rank of links

We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on meridional rank are obtained via Coxeter quotients of the groups of link complements. Matching upper bounds on bridge number are found using the Wirtinger numbers of link diagrams, a combinatorial tool developed by the authors.

The special rank of virtual knot groups

In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications 24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The [Formula: see text]-orbifold group of a link, Math. Z. 200 (1989) 187–208], we prove that classical knots with special rank [Formula: see text] are [Formula: see text]-bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered.

Wirtinger numbers for virtual links

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh’s Tube map to these knots we can obtain nontrivial weakly superslice links.

Symmetric quotients of knot groups and a filtration of the Gordian graph

We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The nth subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.

GEOTECHNICAL CHARACTERISTICS OF THE TERRAIN AND CALCULATION OF BEARING CAPACITY FOR BRIDGE No. 3 OF MOTORWAY TARCIN – KONJIC

In paper is given overview on geotechnical characteristics of the terrain on location ofa bridge number 3, on highway Tarcin – Konjic. Complexity of geological structure,in near surface where it consists from sediments of weathering crust to depth ofaround 30,0 m, as well as in substrate that consists of sediments of clayey debris,determined the way of foundation.On software package Geo 5 is done a semi-analytical approach for determining theload capacity of the pile and extent of subsidence in relation to given load. Pile ismodeled with beam elements where is observed behavior of surrounding terrainaccording to Winkler-Pasternak modele, and shearing force to the contact of pile andsoil are determined based on Mohr-Coulombo criteria. Changeable characteristics onthe terrain demanded analysis of every pile for which are given propositions forfoundation and allowed load.