arXiv subjects

Ferihe Atalan

Publications and source records attributed to Ferihe Atalan.

6 recordsLinked to original sources

arXiv2026

Recognising conjugacy classes of Dehn twists on $\mathbb D_3$ via Dynnikov coordinates

We describe in terms of Dynnikov coordinates the three orbits of the pure mapping class group action on the set of essential curves in a 3-punctured disc $\mathbb D_3$. For any essential curve $\gamma\subset\mathbb D_3$ we present an efficient algorithm to untwist $\gamma$ into one of the 3 basic representatives of the orbits. In turn, we give transformation formulas between the Dynnikov and torus $\mathbb Z^2$-coordinates for multicurves. Besides proving minimality of the algorithm, we give an explicit minimal length formula in terms of the associated even continued fractions.

math.GT↗

arXiv2025

Black holes in the Dynnikov Coordinate Plane

This work presents an application of Dynnikov coordinates in geometric group theory. We describe the orbits and dynamics of the action of Dehn twists $t_c$ and $t_d$ in the Dynnikov coordinate plane for a thrice-punctured disc $M$, where $c$ and $d$ are simple closed curves with Dynnikov coordinates $(0,1)$ and $(0,-1)$, respectively. This action has an interesting geometric meaning as a piecewise linear $\mathbb{Z}^{2}$-automorphism preserving the shape of the linearity border fan.

math.GT↗

arXiv2015

An algebraic characterization of a Dehn twist for nonorientable surfaces

Let $N_g^k$ be a nonorientable surface of genus \ $g\geq 5$ \ with \ $k$-punctures. In this note, we will give an algebraic characterization of a Dehn twist about a simple closed curve on $N_g^k$. Along the way, we will fill some little gaps in the proofs of some theorems in \cite{A} and \cite{I1} giving algebraic characterizations of Dehn twists about separating simple closed curves. Indeed, our results will give an algebraic characterization for the topological type of Dehn twists about separating simple closed curves.

math.GT↗

arXiv2014

Automorphisms of the mapping class group of a nonorientable surface

Let $S$ be a nonorientable surface of genus $g\ge 5$ with $n\ge 0$ punctures, and $\Mcg(S)$ its mapping class group. We define the complexity of $S$ to be the maximum rank of a free abelian subgroup of $\Mcg(S)$. Suppose that $S_1$ and $S_2$ are two such surfaces of the same complexity. We prove that every isomorphism $\Mcg(S_1)\to\Mcg(S_2)$ is induced by a diffeomorphism $S_1\to S_2$. This is an analogue of Ivanov's theorem on automorphisms of the mapping class groups of an orientable surface, and also an extension and improvement of the first author's previous result.

math.GT↗

arXiv2009

Outer Automorphisms of Mapping Class Groups of Nonorientable Surfaces

Let Ng be the connected closed nonorientable surface of genus g >= 5 and Mod(Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod(Ng) is either trivial or Z if g is odd, and injects into the mapping class group of sphere with four holes if g is even.

math.GT↗

arXiv2008

The Number of Pseudo-Anosov Elements in the Mapping Class Group of a Four-Holed Sphere

We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the identity tends to one as the radius of the ball tends to infinity.

math.GT↗