卷 214, 编号 5 (2016)

In the present paper we consider local minimal and extremal networks in normed spaces. It is well known that in the case of the Euclidean space these two classes coincide and the length of a local minimal network can be found by using only the coordinates of boundary vertices and the directions of boundary edges (the Maxwell formula). Moreover, as was shown by Ivanov and Tuzhilin [15], the length of a local minimal network in the Euclidean space can be found by using the coordinates of boundary vertices and the structure of the network. In the case of an arbitrary norm there are local minimal networks that are not extremal networks, and an analogue of the formula mentioned above is only true for extremal networks; this is the main result of the paper. Moreover, we generalize the Maxwell formula for the case of extremal networks in normed spaces and give an explicit construction of norming functionals used in the formula for several normed spaces.

The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link.

In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number.

In the present paper from Derevnin–Mednykh’s formula we obtain integral formulas for the volume of an arbitrary hyperbolic octahedron with mmm- and 2|m-symmetry in terms of dihedral angles.

We consider functorial maps and weak parities that are two equivalent descriptions of one object. Functorial maps allow one to transform knots and extend knot invariants with these transformations. We introduce maximal weak parity and calculate it for knots in a given closed oriented surface. The weak parity induce a projection from virtual knots onto classical ones.

In the present paper, we construct an invariant of knots in the thickened sphere with g handles dependent on 2g + 3 variables. In the construction of the invariant we use the Wirtinger presentation of the knot group and the concept of parity introduced by Manturov [9]. In the present paper, we also consider examples of knots in the thickened torus considered in [2] such that their nonequivalence is proved by using the constructed polynomial.