arXiv subjects

Sreekrishna Palaparthi

Publications and source records attributed to Sreekrishna Palaparthi.

3 recordsLinked to original sources

arXiv2023

Distance $4$ curves on closed surfaces of arbitrary genus

Let $S_g$ denote a closed, orientable surface of genus $g \geq 2$ and $\mathcal{C}(S_g)$ be the associated curve complex. The mapping class group of $S_g$, $Mod(S_g)$ acts on $\mathcal{C}(S_g)$ by isometries. Since Dehn twists about certain curves generate $Mod(S_g)$, one can ask how Dehn twists move specific vertices in $\mathcal{C}(S_g)$ away from themselves. We show that if two curves represent vertices at a distance $3$ in $\mathcal{C}(S_g)$ then the Dehn twist of one curve about another yields two vertices at distance $4$. This produces many tractable examples of distance $4$ vertices in $\mathcal{C}(S_g)$. We also show that the minimum intersection number of any two curves at a distance $4$ on $S_g$ is at most $(2g-1)^2$.

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arXiv2022

Reducing spheres of genus-2 Heegaard splitting of $S^3$

The Goeritz group of the standard genus-g Heegaard splitting of the three sphere, $G_g$, acts on the space of isotopy classes of reducing spheres for this Heegaard splitting. Scharlemann MR2199366 (2007c:57020) uses this action to prove that $G_2$ is finitely generated. In this article, we give an algorithm to construct any reducing sphere from a standard reducing sphere for a genus-2 Heegaard splitting of the $S^3$. Using this we give an alternate proof of the finite generation of $G_2$ assuming the finite generation of the stabilizer of the standard reducing sphere.

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arXiv2019

Describing elements of the genus-2 Goeritz group of $S^3$

In this article we present a finite generating set $G_2$ of $\mathcal{H}_2$, the genus-2 Goeritz group of $S^3$, in terms of Dehn twists about certain simple closed curves on the standard Heegaard surface. We present an algorithm that describes an element $ψ\in\mathcal{H}_2$ as a word in the alphabet of $G_2$ in a certain format. Using a complexity measure defined on reducing spheres, we show that such a description of $ψ$ is unique.

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