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
- 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 \( n. \)
- 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 \( g \geq 2. \) 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.
- 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.
- Panda S., Palaparthi S., Describing elements in the genus-2 Goeritz group of \( S^3 \) (under preparation).
