We develop an algebraic representation for (1, 1)-knots using the mapping class group of the twice punctured torus MCG2(T ). We prove that every (1, 1)-knot in a lens space L(p, q) can be represented by the composition of an element of a certain rank two free subgroup of MCG2(T ) with a standard element only depending on the ambient space. As notable examples, we obtain a representation of this...

In this paper we show that periodic Takahashi 3-manifolds are cyclic coverings of the connected sum of two lens spaces (possibly cyclic coverings of S), branched over knots. When the base space is a 3-sphere, we prove that the associated branching set is a two-bridge knot of genus one, and we determine its type. Moreover, a geometric cyclic presentation for the fundamental groups of these manif...

We prove that octants are cover-decomposable, i.e., any 12-fold covering of any subset of the space with a finite number of translates of a given octant can be decomposed into two coverings. As a corollary, we obtain that any 12-fold covering of any subset of the plane with a finite number of homothetic copies of a given triangle can be decomposed into two coverings. We also show that any 12-fo...

We prove that two automata with multiplicity in Z are equivalent, i.e. define the same rational series, if and only if there is a sequence of Z-coverings, co-Z-coverings, and circulations of −1, which transforms one automaton into the other. Moreover, the construction of these transformations is effective. This is obtained by combining two results: the first one relates coverings to conjugacy o...

A planar set P is said to be cover-decomposable if there is a constant k = k(P ) such that every k-fold covering of the plane with translates of P can be decomposed into two coverings. It is known that open convex polygons are cover-decomposable. Here we show that closed, centrally symmetric convex polygons are also cover-decomposable. We also show that an infinite-fold covering of the plane wi...

In this note we improve our upper bound given in [7] by showing that every 9-fold covering of a point set in R by finitely many translates of an octant decomposes into two coverings, and our lower bound by a construction for a 4-fold covering that does not decompose into two coverings. We also prove that certain dynamic interval coloring problems are equivalent to the above question. The same b...

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...