arXiv subjects

Bidyut Sanki

Publications and source records attributed to Bidyut Sanki.

15 recordsLinked to original sources

arXiv2026

Systole Increasing Deformations to Maximal Translation Surfaces

A unit-area translation surface is called \emph{maximal} if it maximizes the length of the shortest saddle connection among all surfaces in the same stratum. We investigate whether a non-maximal translation surface can be continuously deformed into a maximal one while the systole increases strictly monotonically. Although such a deformation does not exist in general due to the existence of local but not global maxima of the systole function, we prove that it exists for square-tiled surfaces and translation surfaces obtained from regular hexagons and regular octagons. In each case, we explicitly construct a continuous deformation to a maximal translation surface along which the systole is strictly increasing. Finally, the preceding construction yields another family of translation surfaces admitting continuous systole-increasing deformations to maximal surfaces.

math.GT↗

arXiv2026

Minimal Filling pair of non orientable surfaces

For $g\ge 3$, let $N_g$ denote the non-orientable surface of genus $g$. In this article, we establish the existence of filling pairs on $N_g$ that intersect minimally by construction using the theory of fat graphs. The mapping class group $\mathrm{Mod}(N_g)$ acts on the set of all such filling pairs. We count $\mathrm{Mod}(N_g)$-orbits of this action by providing both lower and upper bounds. Furthermore, we show that both bounds grow super-exponentially with $g$ using graph cohomology. Also, we investigate the lengths of minimally intersecting filling pairs on hyperbolic non-orientable surfaces $X$ in moduli space $\mathcal{M}_g$ of $N_g$. We define a function $\mathcal{F}_g:\mathcal{M}_g\to\mathbb{R}_{>0}$, where for $X\in \mathcal{M}_g$, the function $\mathcal{F}_g(X)$ is the shortest total length of a minimally intersecting filling pair on $X$. We determine its minimum $m_g$ and show that the set of minimizers is in bijection with the $\mathrm{Mod}(N_g)$-orbits of minimally intersecting filling pairs. We further extend $\mathcal{F}_g$ to $\mathcal{Y}_g$, defined by minimizing the length over all filling pairs, and show that $\mathcal{Y}_g$ attains the same minimum value as $\mathcal{F}_g$.

math.GT↗

arXiv2026

Length minimization of filling pairs on hyperbolic surfaces

A filling pair $(α, β)$ of a surface $S_g$ is a pair of simple closed curves in minimal position such that the complement of $α\cupβ$ in $S_g$ is a disjoint union of topological disks. A filling pair is said to be minimally intersecting if the number of intersections between them, or equivalently, the number of complementary disks, is minimal among all filling pairs of $S_g$. For surfaces of genus $g \geq 3$, minimal filling pairs are well understood, whereas in genus two, such a pair divides the surface into exactly two disks. In this paper, we classify all minimal filling pairs up to the action of the mapping class group in genus two and determine the length of the shortest minimal filling pair.

math.GT↗

arXiv2025

Systolic embedding of graphs on translation surfaces

An embedding of a graph on a translation surface is said to be \emph{systolic} if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation surface. The embedding is said to be \emph{cellular} (respectively \emph{essential}) if each complementary region is a topological disk (respectively not a topological disk). In this article, we prove that any finite graph admits an essential-systolic embedding on a translation surface and estimate the genera of such surfaces. For a wedge $\Sigma_n$ of $n$ circles, $n\geq2$, we investigate that $\Sigma_n$ admits cellular-systolic embedding on a translation surface and compute the minimum and maximum genera of such surfaces. Finally, we have identified another rich collection of graphs with more than one vertex that also admit cellular-sytolic embedding on translation surfaces.

math.GT↗

arXiv2025

Bolza-like surfaces in the Thurston set

A surface in the Teichm\"uller space, where the systole function admits its maximum, is called a maximal surface. For genus two, a unique maximal surface exists, which is called the Bolza surface, whose systolic geodesics give a triangulation of the surface. We define a surface as Bolza-like if its systolic geodesics decompose the surface into $(p, q, r)$-triangles for some integers $p,q,r$. In this article, we will provide a construction of Bolza-like surfaces for infinitely many genera $g\geq 9$. Next, we see an intriguing application of Bolza-like surfaces. In particular, we construct global maximal surfaces using these Bolza-like surfaces. Furthermore, we study a symmetric property satisfied by the systolic geodesics of our Bolza-like surfaces. We show that any simple closed geodesic intersects the systolic geodesics at an even number of points.

math.GT↗

arXiv2024

Length of Filling Pairs on Punctured Surface

A pair $(\alpha, \beta)$ of simple closed curves on a surface $S_{g,n}$ of genus $g$ and with $n$ punctures is called a filling pair if the complement of the union of the curves is a disjoint union of topological disks and once punctured disks. In this article, we study the length of filling pairs on once-punctured hyperbolic surfaces. In particular, we find a lower bound of the length of filling pairs which depends only on the topology of the surface.

math.GT↗

arXiv2024

Filling with separating curves

A pair $(α, β)$ of simple closed curves on a closed and orientable surface $S_g$ of genus $g$ is called a filling pair if the complement is a disjoint union of topological disks. If $α$ is separating, then we call it as separating filling pair. In this article, we find a necessary and sufficient condition for the existence of a separating filling pair on $S_g$ with exactly two complementary disks. We study the combinatorics of the action of the mapping class group $\M$ on the set of such filling pairs. Furthermore, we construct a Morse function $\mathcal{F}_g$ on the moduli space $\mathcal{M}_g$ which, for a given hyperbolic surface $X$, outputs the length of shortest such filling pair with respect to the metric in $X$. We show that the cardinality of the set of global minima of the function $\mathcal{F}_g$ is the same as the number of $\M$-orbits of such filling pairs.

math.GT↗

arXiv2023

Complexity in Bolza surface

A surface in the Teichm\"uller space, where the systole function attains its maximum, is called a maximal surface. For genus two there exists a unique maximal surface which is called the Bolza surface. In this article, we study the complexity of the set of systolic geodesics on the Bolza surface. We show that any non-systolic geodesic intersects the systolic geodesics in $2n$ points, where $n\geq 5$. Furthermore, we show that there are $12$ second systolic geodesics on the Bolza surface and they form a triangulation of the surface.

math.DG↗

arXiv2021

Isoperimetric Inequality for Disconnected Regions

The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic geometry as well. In this paper, we generalize the discrete isoperimetric inequality to disconnected regions, i.e. we allow the area to be split between regions. We give necessary and sufficient conditions for the result (in Euclidean, spherical and hyperbolic geometry) to hold for multiple $n$-gons whose areas add up.

math.GT↗

arXiv2020

A conjecture on the lengths of filling pairs

A pair $(α, β)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $α\cupβ$ in $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In \cite{Aou}, Aougab-Huang conjectured that the length of any filling pair on $M$ is at least $\frac{m_{g}}{2}$, where $m_{g}$ is the perimeter of the regular right-angled hyperbolic $\left(8g-4\right)$-gon. In this paper, we prove a generalized isoperimetric inequality for disconnected regions and we prove the Aougab-Huang conjecture as a corollary.

math.GT↗

arXiv2018

Embedding of metric graphs on hyperbolic surfaces

An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_e(G)$ of $(G, d)$ is the lowest genus of a surface on which such an embedding is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ admits such an embedding (possibly after a rescaling of $d$) on a surface of genus $g$. Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_e^{\max}(G)$ of $(G, d)$ is the largest genus of a surface on which the graph is minimally embedded. Finally, we describe a method explicitly for an essential embedding of $(G, d)$, where $g_e(G)$ and $g_e^{\max}(G)$ are realized.

math.GT↗

arXiv2018

Filling systems on surfaces

Let $F_g$ be a closed orientable surface of genus $g$. A set $Ω= \{ γ_1, \dots, γ_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a \emph{filling system} or simply a \emph{filling} of $F_g$, if $F_g\setminus Ω$ is a union of $b$ topological discs for some $b\geq 1$. A filling system is called \emph{minimal}, if $b=1$. The \emph{size} of a filling is defined as the number of its elements. We prove that the maximum size of a filling of $F_g$ with $b$ complementary discs is $2g+b-1$. Next, we show that for $g\geq 2, b\geq 1\text{ with }(g,b)\neq (2,1)$ (resp. $(g,b)=(2,1)$) and for each $2\leq s\leq 2g+b-1$ (resp. $3\leq s\leq 2g+b-1$), there exists a filling of $F_g$ of size $s$ with $b$ complementary discs. Furthermore, we study geometric intersection number of curves in a minimal filling. For $g\geq 2$, we show that for a minimal filling $Ω$ of size $s$, the \emph{geometric intersection numbers} satisfy $\max \left\lbrace i(γ_i, γ_j)| i\neq j\right\rbrace\leq 2g-s+1$, and for each such $s$ there exists a minimal filling $Ω=\left\lbrace γ_1, \dots, γ_s \right\rbrace$ such that $\max\left\lbrace i(γ_i, γ_j) | i\neq j\right\rbrace = 2g-s+1$.

math.GT↗

arXiv2017

Geometric realizations of cyclic actions on surfaces

Let $ \text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$, and let $f\in \text{Mod}(S_g)$ be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on $S_g$ that realizes $f$ as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of $ \text{Mod}(S_g)$. Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation $Ψ: \text{Mod}(S_g) \to \text{Sp}(2g; \mathbb{Z})$.

math.GT↗

arXiv2017

Filling of closed Surfaces

Let $F_g$ denote a closed oriented surface of genus $g$. A set of simple closed curves is called a filling of $F_g$ if its complement is a disjoint union of discs. The mapping class group $\text{Mod}(F_g)$ of genus $g$ acts on the set of fillings of $F_g$. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of $F_g$ are in the same $\text{Mod}(F_g)$-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of $F_2$ whose complement is a single disc (i.e., a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of $\text{Mod}(F_2)$. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of $F_2$ is two. Finally, given positive integers $g$ and $k$ with $(g, k)\neq (2, 1)$, we construct a filling pair of $F_g$ such that the complement is a union of $k$ topological discs.

math.GT↗

arXiv2017

Graphs of Systoles on hyperbolic surfaces

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible). There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient. It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

math.GT↗