Sucharit Sarkar
সুচরিত সরকার
ʃuʧorit̪ ʃɔrkar

Associate Professor
Department of Mathematics, Topology group
University of California at Los Angeles

Office 6909 MS
Office Phone
Department of Mathematics
University of California at Los Angeles
520 Portola Plaza
Box 95155
Los Angeles, CA 90095-1555

Current courses
 Fall Winter Spring Summer 2020-21: 100 101 236 - -
Old courses
 2019-20: 100 101 - 225C - 2018-19: 100 120A 225C - 2017-18: 100 225B 32A - 2016-17: 120A 285F 225C 31B

CV

Journals

Publication and preprints
(Manolescu-Marengon-Sarkar-Willis) A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds.
ArXiv
We generalize Rasmussen's s-invariant for certain links inside connected sums of S^1\times S^2's, and prove adjunction inequality for s-invariant in CP^2. As a corollary, we prove that the Freedman-Gompf-Morrison-Walker strategy cannot be used to prove that Gluck twists are not diffeomorphic to the 4-ball.
(Lipshitz-Sarkar) Khovanov homology also detects split links.
ArXiv
We prove that the module structure on Khovanov homology detects split links.
(Lawson-Lipshitz-Sarkar) Chen-Khovanov spectra for tangles. Michigan Math. J.
ArXiv
We define stable homotopy refinements of Chen-Khovanov algebras and tangle invariants.
(Sarkar) Ribbon distance and Khovanov homology. Algebr. Geom. Topol.
We use orders of certain torsion elements in Khovanov homology to produce lower bounds on complexities of ribbon concordances between knots.
(Hendricks-Lipshitz-Sarkar) Correction to the paper "A flexible construction of equivariant Floer homology and applications". J. Topol.
We correct a mistake regarding almost complex structures on Hilbert schemes of points in surfaces in arXiv:1510.02449 and give an alternate proof of the invariance of equivariant symplectic Khovanov homology.
(Lipshitz-Sarkar) Stable homotopy refinements and Khovanov homology.
These are slides from a talk that we gave at the 2018 ICM.
We produce a spatial refinement of odd Khovanov homology, similar in spirit to the spatial refinement of even Khovanov homology, but using functors from a signed version of the Burnside category.
(Lipshitz-Sarkar) Spatial refinements and Khovanov homology. Proc. of the ICM
We review the construction and context of a stable homotopy refinement of Khovanov homology.
(Lawson-Lipshitz-Sarkar) Khovanov spectra for tangles.
ArXiv
We define stable homotopy refinements of Khovanov's arc algebras and tangle invariants.
(Sarkar) Phutball draws. Games of No Chance 5
In this short note, we exhibit a draw in the game of Philosophers Phutball. We construct a position on a 12 x 10 Phutball board from where either player has a drawing strategy, and then generalize it to an m x n board with m-2 >= n >= 10.
(Hendricks-Lipshitz-Sarkar) A simplicial construction of G-equivariant Floer homology. Proc. London Math. Soc.
ArXiv
For G a Lie group acting on a symplectic manifold preserving a pair of Lagrangians L0, L1, under certain hypotheses not including equivariant transversality we construct a G-equivariant Floer cohomology of L0 and L1.
(Baldwin-Levine-Sarkar) Khovanov homology and knot Floer homology for pointed links. J. Knot Theory Ramifications
A well-known conjecture states that for any l-component link L in S3, the rank of the knot Floer homology of L (over any field) is less than or equal to 2^(l-1) times the rank of the reduced Khovanov homology of L. In this paper, we describe a framework that might be used to prove this conjecture.
(Hendricks-Lipshitz-Sarkar) A flexible construction of equivariant Floer homology and applications. J. Topol.
In this paper we give a new construction of equivariant Floer cohomology with respect to a finite group action on a symplectic manifold and use it to prove some invariance properties of these spectral sequences, and prove it agrees with some existing spectral sequences.
(Lawson-Lipshitz-Sarkar) The cube and the Burnside category. Contemp. Math.
In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between two combinatorial 2-categories, modulo a notion of stable equivalence. We also develop some general properties of such functors.
(Lawson-Lipshitz-Sarkar) Khovanov homotopy type, Burnside category, and products. Geom. Topol.
ArXiv
In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in Lipshitz-Sarkar and Hu-Kriz-Kriz and, as a corollary, that those two constructions give equivalent spaces.
(Sarkar-Seed-Szabo) A perturbation of the geometric spectral sequence in Khovanov homology. Quantum Topol.
We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new invariant, and introduce a family of s-invariants from the new theory in the same spirit as Rasmussen's s-invariant.
(Lipshitz-Ng-Sarkar) On transverse invariants from Khovanov homology. Quantum Topol.
In this paper, we give two refinements of Plamenevskaya's invariant of transverse knots, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these refinements is invariant under negative flypes and SZ moves.
(Lipshitz-Sarkar) A refinement of Rasmussen's s-invariant. Duke Math. J.
In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology, which we use to construct refinements of Rasmussen's slice genus bound s. We show that in the case of the Steenrod square Sq^2 our refinement is strictly stronger than s.
(Lipshitz-Sarkar) A Steenrod square on Khovanov homology. J. Topol.
In a previous paper, we defined a space-level version X(L) of Khovanov homology. This induces an action of the Steenrod algebra on Khovanov homology. In this paper, we describe the first interesting operation, Sq^2:Kh^{i,j}(L) -> Kh^{i+2,j}(L). We compute this operation for all links up to 11 crossings; this, in turn, determines the stable homotopy type of X(L) for all such links.
(Everitt-Lipshitz-Sarkar-Turner) Khovanov homotopy types and the Dold-Thom functor. Homology Homotopy Appl.
We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. By using the Dold-Thom functor it can therefore be obtained from the Khovanov homotopy type constructed by Lipshitz and Sarkar.
(Lipshitz-Sarkar) A Khovanov stable homotopy type. J. Amer. Math. Soc.
Given a link diagram L we construct spectra X^j(L) so that the Khovanov homology Kh^{i,j}(L) is isomorphic to the (reduced) singular cohomology H^i(X^j(L)). The construction of X^j(L) is combinatorial and explicit. We prove that the homotopy type of X^j(L) depends only on the isotopy class of the corresponding link.
(Sarkar) Moving basepoints and the induced automorphisms of link Floer homology. Algebr. Geom. Topol.
Given an l-component pointed oriented link (L,p) in an oriented three-manifold Y, one can construct its link Floer chain complex CFL(Y,L,p) over the polynomial ring F_2[U_1,...,U_l]. Moving the basepoint p_i in the link component L_i once around induces an automorphism of CFL(Y,L,p). In this paper, we study an automorphism (a possibly different one) of CFL(Y,L,p) defined explicitly in terms of holomorphic disks; for links in S^3, we show that these two automorphisms are the same.
(Sarkar) Grid diagrams and the Ozsvath-Szabo tau-invariant. Math. Res. Lett.
We use grid diagrams to investigate the Ozsvath-Szabo concordance invariant tau, and to prove that |tau(K_1)-tau(K_2)|<=g, whenever there is a genus g knot cobordism joining K_1 to K_2. This leads to an entirely grid diagram-based proof of Kronheimer-Mrowka's theorem, formerly known as the Milnor conjecture.
(Sarkar) A note on sign conventions in link Floer homology. Quantum Topol.
Grid diagrams can be used to compute knot Floer homology of links in S^3 over F_2. There are 2^(|L|-1) versions of link Floer homology over Z, and we construct 2^(|L|-1) sign refinements of the grid chain complex and show that the sign refined complexes still compute the sign refined link Floer homology.
(Sarkar) Topics in Heegaard Floer homology.
This is the my PhD thesis, as submitted to the Princeton University.
(Sarkar) Grid diagrams and shellability. Homology Homotopy Appl.
We explore a somewhat unexpected connection between knot Floer homology and shellable posets, via grid diagrams. Given a grid presentation of a knot K inside S^3, we define a poset which has an associated chain complex whose homology is the knot Floer homology of K, and whose each closed interval is shellable.
(Hedden-Juhasz-Sarkar) On sutured Floer homology and the equivalence of Seifert surfaces. Algebr. Geom. Topol.
We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the "top term" of the knot Floer homology, which they contain.
(Sarkar) Maslov index formulas for Whitney n-gons. J. Symplectic Geom.
In this short article, we find an explicit formula for Maslov index of Whitney n-gons joining intersections points of n half-dimensional tori in the symmetric product of a surface. The method also yields a formula for the intersection number of such an n-gon with the fat diagonal in the symmetric product.
(Sarkar-Wang) An algorithm for computing some Heegaard Floer homologies. Ann. of Math.
In this paper, we give an algorithm to compute the hat version of the Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtrations coming from a null-homologous link in a three-manifold.
(Manolescu-Ozsvath-Sarkar) A combinatorial description of knot Floer homology. Ann. of Math.
Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.
(Sarkar) Commutators and squares in free groups. Algebr. Geom. Topol.
Let F_2 be the free group generated by x and y. In this article, we prove that the commutator of x^m and y^n is a product of two squares if and only if mn is even.
(Sarkar) Some properties of subgroup counting function.
We compute the number of subgroups of certain free products of cyclic groups of a fixed finite index modulo certain primes. (This was an undergraduate project.)

Seminars
• The Joint LA Topology Seminar
(You may add the events to your own calendar using these ICAL links: Topology Seminar, Joint Topology Seminar; see for instance this how-to.)

Workshops

Programs
• Analyse Heegaard diagrams: hf-hat
• Compute some Steenrod squares on the Khovanov homology: KhovanovSteenrod
• Compute some new S invariants from the Khovanov homology: newSinvariants
• A modified AMSalpha bibliography style file: myalpha
• Phonetic typing in Bengali, Devnagari on GNU+Linux: Rabindra
• Phonetic typing in Bengali on Android: RabindraKeyboard
• Cross-platform phonetic typing in Bengali: Website

My mathematical genealogy; and as a graph.
My support system, Pami Mukherjee and Nirajana Sarkar; another graph.