We describe the most efficient solutions to the word problem of Artin’s braid group known so far, i.e., in other words, the most efficient solutions to the braid isotopy problem, including the Dynnikov method, which could be especially suitable for cryptographical applications. Most results appear in literature; however, some results about the greedy normal form and the symmetric normal form an...

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be presented. The results are based on joint works of the author with his collabor...

This paper introduced a condition called $\mathcal{R}$-condition under which $(r,s,t)$-inverse quasigroups are universal. Middle isotopic loops, satisfying the and possessing trivial set of $r$-weak inverse permutations were shown to be isomorphic; isotopy-isomorphy for loops. Isotopy-isomorphy loops was generally characterized. With $\mathcal{R}$-condition, it that positive integers $r$, $s$ $...

Let Bn denote the classical braid group on n strands and let the mixed braid group Bm,n be the subgroup of Bm+n comprising braids for which the first m strands form the identity braid. Let Bm,∞ = ∪nBm,n. We will describe explicit algebraic moves on Bm,∞ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented ...

We develop a new approach to the classical problem on isotopy classification of embeddings of manifolds into Euclidean spaces. This approach involves studying of a new embedding invariant, of almost-embeddings and of smoothing, as well as explicit constructions of embeddings. Using this approach we obtain complete concrete classification results below the metastable dimension range, i.e. where ...

To each rail knotoid we associate two unoriented knots along with their oriented counterparts, thus deriving invariants for knotoids based on these associations. We then translate them to of isotopy arcs.

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. study the corresponding isotopy problem, with focus rational irreducible (rational cuspidal curves) projective plane. prove that every such is isotopic to degrees up five, and for one singularity link torus knot. Classification results classe...