2019
Том 71
№ 9

# Plakhta L. P.

Articles: 3
Article (Ukrainian)

### Invariants of knots, surfaces in R 3, and foliations

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1239–1252

We give a survey of some known results related to combinatorial and geometric properties of finite-order invariants of knots in a three-dimensional space. We study the relationship between Vassiliev invariants and some classical numerical invariants of knots and point out the role of surfaces in the investigation of these invariants. We also consider combinatorial and geometric properties of essential tori in standard position in closed braid complements by using the braid foliation technique developed by Birman, Menasco, and other authors. We study the reductions of link diagrams in the context of finding the braid index of links.

Brief Communications (Ukrainian)

### Restrictions on free actions of the alternating group $A_6$ on products of spheres

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1431-1434

We prove that the alternating group $A_6$ cannot freely act on $(S^n)^5$ We give an example of free action of the alternating group $A_4$ on $(S^n)^3$.

Article (Ukrainian)

### On the set of singular points of the actions of finite groups on $(S^n)^k$

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1233-1237

Realizations of integral $D_3$-modules of rank 2 on $(S^n)^k$ for the dihedral groups $D_3$ are studied. Cohomologies of the sets of the singular points of the actions of the semidirect products $ℤ / pXℤ / q$ and the quaternion groups $Q$ on $(S^n)^k$ are investigated.