My broad area of research is geometric topology, in particular I work in low-dimensional topology and combinatorial topology. My primary focus is on triangulations, spatial graphs, knots, hyperbolic structures and Heegaard splittings of 3-dimensional manifolds. Below are some of the areas that I have been working on:

• Manifold recognition Triangulations of a manifold allow us to use combinatorial algorithms to resolve problems in geometric topology. Any two PL-triangulations of a PL-manifold are related by a finite sequence of triangulations each obtained from the previous one by a finite number of local combinatorial changes called bistellar moves. An explicit bound on the number of such moves needed in this bistellar sequence leads to a recognition algorithm for PL-manifolds. The geometrisation theorem of Thurston-Perelman says that all 3-dimensional manifolds can be split into pieces which are either hyperbolic or have the structure of a circle bundle over an orbifold, called Seifert fibered spaces.
  
  For geometric triangulations of hyperbolic, spherical and Euclidean n-dimensional manifolds, we have obtained an explicit bound on the length of this bistellar sequence. Related publication: An upper bound on Pachner moves relating geometric triangulations (with my student Advait Phanse).
  
  We have also obtained such a bound for a bistellar sequence between ideal geometric triangulations of cusped hyperbolic manifolds, which leads to an effective algorithm for hyperbolic knot equivalence. The bound is in terms of the number of tetrahedra along with an upper bound on the length of edges in the compact case and a lower bound on the dihedral angles in the cusped case. We also obtain a lower bound on systole lengths of cusped hyperbolic manifolds in terms of the number of tetrahedra and lower dihedral angle bound. Related publication: Bounds on Pachner moves and systoles of cusped 3-manifolds (with my student Sriram Raghunath).
  
  Additionally, we have shown that it is possible to construct this bistellar sequence entirely through triangulations that remain geometric. Related publication: Geometric bistellar moves relate geometric triangulations (with my student Advait Phanse).
  
  
• Normal surfaces in triangulated 3-manifolds A surface embedded in a triangulated 3-manifold is called normal if it intersects each tetrahedron of the triangulation in triangular or quadrilateral disks. Normal surfaces allow us to algorithmically resolve questions about interesting surfaces in 3-manifolds. We have given a lower bound on the Euler characteristic of a normal surface, a topological invariant, in terms of the number of normal quadrilaterals in its embedding. Related publication: Euler characteristic and quadrilaterals of normal surfaces.
  
  Incompressible surfaces are an interesting class of embedded surfaces along which a 3-manifold can be cut and simplified. Haken had proved that when an incompressible surface is isotoped to have least PL-area with respect to a given triangulation, then the surface is in normal form. We have proved a converse of this result, i.e., we have shown that if with respect to every triangulation, a least PL-area representative of the given surface is normal, then the surface must be incompressible. Related publication: Incompressibility and normal minimal surfaces.
  
  And lastly, we have interpreted a normal surface in terms of the homology of a certain chain complex. This allowed us to give a simpler proof of Casson-Rubinstein-Tollefson’s result that quadrilateral coordinates are sufficient to determine a normal surface (up to vertex-linking spheres). Related publication: A chain complex and Quadrilaterals for normal surfaces (with Siddhartha Gadgil).
  
  
• Seifert Fibered spaces For Seifert fibered manifolds, we have introduced the notion of prism complexes in place of simplicial complexes; and shown that while every 3-manifold has a prism complex, it has a special prism complex if and only if it is a Seifert fibered space. Related publication: Prism complexes (with my student Ramya Nair).
  
  Incompressible surfaces are well-studied in orientable Seifert fiber spaces. But not much work has been done in the non-orientable case, where the model neighbourhood around a fiber can be either a fibered solid torus or a fibered solid Klein bottle, and hence the singular fibers may not be isolated. Extending the work of Frohman and Rannard to Seifert fiber spaces with such singular surfaces, we have shown that incompressible surfaces in such manifolds are of one of two types: pseudo-horizontal or pseudo-vertical. Related publication: Essential surfaces in Seifert fiber spaces with singular surfaces (with my student Ramya Nair).
  
  
• Taut Foliations A foliation of a 3-manifold is a decomposition into 2-dimensional injectively immersed submanifolds that are locally parallel. Taut foliations are a special kind of foliation with close links to contact structures and open book decompositions. Roberts had shown that given a surface bundle over a circle with connected boundary, the surface fibers can be perturbed to taut foliations that realise all rational boundary slopes in some neighbourhood of the slope of the fiber. We have extended her result to the multiple boundary case using the idea of laminar branched surfaces developed by Tao Li. Related publication: Taut foliations in surface bundles with multiple boundary components (with Rachel Roberts).
  
  
• Heegaard Splittings A Heegaard splitting is a way to cut a 3-manifold along an embedded surface into simpler pieces called handlebodies. Gabai and Colding gave an effective version of Li’s theorem that there are only finitely many irreducible splittings of a non-Haken hyperbolic manifold. They end with an open question for the structure of irreducible splittings of Haken manifolds. We have extended their result and resolved their question for strongly-irreducible Heegaard splittings of all closed hyperbolic 3-manifolds. Our result says that there exist finitely many strongly-irreducible splittings S_i and finitely many incompressible surfaces K_j such that every such splitting is a Haken sum S_i + Σ n_j K_j (up to one-sided associates). Related publication: Strongly irreducible Heegaard splittings of hyperbolic 3-manifolds.
  
  
• Tilings Inspired by the circle packing theorem of Koebe-Andreev-Thurston, Feng Lou conjectured that there is a unique tiling of the plane by squares (up to scaling and sliding) with contacts graph the 1-skeleton of the hexagonal triangulation of the plane, i.e., each square has exactly six neighbours. We have constructed families of counter-examples of a similar statement for tilings of the punctured plane. This project is joint work with Chaim Zohar and Maria Trnkova and is currently on hold.
  
  
• Spatial Graphs Spatial graphs is the study of graphs embedded in R^3. We have obtained a linear algebraic test to check if two 3-regular spatial graphs are isomorphic by using the writhes of the circuits in the graph. Related publication: Writhe invariants of 3-regular spatial graphs (with Stefan Friedl and José Pedro Quintanilha).
  
  
• Connecting higher-dimensional manifold triangulations via flips Benedetti-Petronio fine-tuned Pachner’s result using the dual view-point of spines to show that any two triangulations of a 3-manifold with the same number of vertices are related by bistellar moves through triangulations with the same number of vertices. Matveev and Amendola generalised this to cusped 3-manifolds. This is beneficial both for defining 3-manifold invariants using triangulations and also for improving efficiency in making censuses of 3-manifolds. We are currently working on extending this result to triangulations of higher dimensional manifolds. This is a joint project with Henry Segerman.


• Connecting essential 3-dimensional manifold triangulations via flips Edges of a triangulated 3-manifold are called inessential if they are trivial loops. Triangulations of a 3-manifold with all edges essential are called essential triangulations. We have shown that any two essential triangulations of a 3-manifold are related by a sequence of bistellar moves through essential triangulations, without introducing or removing any vertices. In particular, ideal essential triangulations of a cusped manifold are related by a sequence of bistellar moves through ideal essential triangulations. These results are proved in the more general setting of L-essential triangulations, i.e., triangulations that are essential with respect to a labelling on the lift of the cusps of the manifold to the universal cover. Given a representation, we also construct an ideal triangulation for which a solution to the Thurston’s gluing equations recovers the representation. These results lead to the invariance of the quantum 1-loop invariant. Related publications: Connecting essential triangulations I: via 2-3 and 0-2 moves and Connecting essential triangulations II: via 2-3 moves only (with Saul Schleimer and Henry Segerman).
  
  

For career and education details see my CV.