Research

Broad Research Area:

• Geometry and Topology

Specific Research Keywords:

• Low dimensional Topology
• Knot Theory
• Hyperbolic 3-manifolds
• Heegaard Splittings of 3-manifolds
• Surface Mapping Class Groups
• 3-manifold Mapping Class Groups
• Curve Complexes
• Teichmuller Theory
• Geometric Group Theory



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Publications:



1. Palaparthi S., Closed Geodesic Lengths in Hyperbolic Link Complements in $S^3$ , International Journal of Pure and Applied Mathematics, Volume 83, No. 1, 2013.

   Summary: Adams and Reid proved that the systole in any hyperbolic knot or link complement in $S^3$ is bounded above by a constant. In this article, we prove that the length of the $n^{th}$ shortest closed geodesic in hyperbolic knot and link complements in $S^3$ are also bounded above by a constant which depends only on $n.$

2. Mahanta K., Palaparthi S., Distance 4 curves on closed surfaces of arbitrary genus , Topology and its Applications, Volume 314, 2022.

   Summary: Let $\alpha$ and $\beta$ be two curves corresponding to two vertices in the curve complex of a closed surface of genus $g\ge 2.$ Let $T_{\beta }(\alpha )$ be the Dehn twist of $\alpha$ about $\beta$. In this article, we show that if $\alpha$ and $\beta$ are a filling pair, then so are $\alpha$ and $T_{\beta }(\alpha )$. Further, if $\alpha$ and $\beta$ are at a distance 3 in the curve graph then $\alpha$ and $T_{\beta }(\alpha )$ are at a distance 4 apart. This produces infinitely many explicit examples of curves at a distance 4 in the curve complex of closed surfaces. Using this we give better upper bounds for the minimal intersection number of any two curves at a distance 4 in the curve complex of such closed surfaces.

3. Palaparthi S., Panda S. Reducing spheres of the genus-2 Heegaard splitting of $S^3$ (accepted for publication in Journal of Topology and Analysis).

   Abstract: 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 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.

4. Panda S., Palaparthi S., Describing elements in the genus-2 Goeritz group of $S^3$ (under preparation).