arXiv subjects
Publications and source records attributed to Y. Belousov.
We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial. As an application, we construct a new family of knots that are topologically slice but not smoothly slice.
In this paper, we introduce a prime factorization of open meanders, articulated through the framework of 2-colored operads. We demonstrate that each open meander can be canonically constructed from building blocks of two types: iterated snakes and irreducible meanders. We find out that iterated snakes allow efficient enumeration, and thus the problem of enumerating meanders reduces to the problem of enumerating irreducible meanders. Additionally, we present some results concerning the asymptotic of meanders of both classes.
For classical knots, there is a concept of (semi)meander diagrams; in this short note we generalize this concept to virtual knots and prove that the classes of meander and semimeander diagrams are universal (this was known for classical knots). We also introduce a new class of invariants for virtual knots -- virtual $k$-arc crossing numbers and we use Manturov projection to show that for all classical knots the virtual $k$-arc crossing number equals to the classical $k$-arc crossing number.
In this article, we provide upper and lower bounds for the growth rate of irreducible meanders. The obtained upper bound implies that the proportion of irreducible meanders among all of the prime meanders of order $n$ approaches $0$ as $n$ approaches infinity.