On polynomial isotopy of knot-types

We have proved that every knot-type ???³ can be uniquely represented by polynomials up to polynomial isotopy i.e. if two polynomial embeddings of ? in ?³ represent the same knot-type, then we can join them by polynomial embeddings.     

On extension of some generalizations of quasiconformal mappings to a boundary

This work is devoted to the investigation of ring Q-homeomorphisms. We formulate conditions for a function Q(x) and the boundary of a domain under which every ring Q-homeomorphism admits a homeomorphic extension to the boundary. For an arbitrary ring Q-homeomorphism f: D → D’ with Q ∈ L ¹(D); we study the problem of the extension of inverse mappings to the boundary. It is proved that an isolated singularity is removable for ring Q-homeomorphisms if Q has finite mean oscillation at a point.     

On the isotopy of Legendrian knots

Let ₀ and ₁ be Legendrian knots which are isotopic as usual knots, and which have the same obvious invariants rot and link. It seems to be an open question whether ₀ and ₁ are isotopic as Legendrian knots. In the paper we give a positive answer to this question for the (rather restricted) class of Legendrian knots with nonintersecting fronts.     

On the extremality of quasiconformal mappings and quasiconformal deformations

Given a family of quasiconformal deformations such that has a uniform bound , the solution of the Löwner-type differential equation

is an -quasiconformal mapping. An open question is to determine, for each fixed , whether the extremality of is equivalent to that of . The note gives this a negative approach in both directions.

     

On some results for a class of meromorphic functions having quasiconformal extension

We consider the class \(\Sigma (p)\) of univalent meromorphic functions f on \({\mathbb D}\) having a simple pole at \(z=p\in [0,1)\) with residue 1. Let \(\Sigma _k(p)\) be the class of functions in \(\Sigma (p)\) which have k-quasiconformal extension to the extended complex plane \({\hat{\mathbb C}}\), where \(0\le k < 1\). We first give a representation formula for functions in this class and using this formula, we derive an asymptotic estimate of the Laurent coefficients for the functions in the class \(\Sigma _k(p)\). Thereafter, we give a sufficient condition for functions in \(\Sigma (p)\) to belong to the class \(\Sigma _k(p).\) Finally, we obtain a sharp distortion result for functions in \(\Sigma (p)\) and as a consequence, we obtain a distortion estimate for functions in \(\Sigma _k(p).\)     

On locally quasiconformal mappings

A homeomorphism w=f(z) of a domain D is called a locally quasiconformal mapping, if for each subdomain D' of D with 'D, the restriction of f(z) on D' is a quasiconformal mapping. We give some conditions for a measurable function μ(z) on the unit disc to be the complex dilatation of a locally quasiconformal mapping f which can be homeomorphically extended to the closed unit disc.     

On mean quasiconformal mappings

The research was partly supported by the International Science Foundation (Grant U 96 000).     

Extremal problems in the theory of quasiconformal mappings

Kemerovo. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 202–204, January–February, 1990.