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In this short note, we exhibit a draw in the game of Philosopher’s 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.

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Based on a lecture series, this is a review of spatial refinements of Khovanov homology and knot Floer homology (via grid diagrams).

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We construct a spectral sequence relating the Khovanov homology of a strongly invertible knot to the annular Khovanov homologies of the two natural quotient knots. Using this spectral sequence, we re-prove that Khovanov homology distinguishes certain slice disks. We also give an analogous spectral sequence for the Heegaard Floer homology of the branched double cover.

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We establish a graded version of Ni's isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that tau=g for each summand if and only if tau=g holds for the Murasugi sum (with tau and g defined appropriately for multi-component links).

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We construct a mixed invariant of non-orientable surfaces from the Lee and Bar-Natan deformations of Khovanov homology and use it to distinguish pairs of surfaces bounded by the same knot, including some exotic examples.

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Given a grid diagram for a knot or link K in S3, we construct a spectrum whose homology is the knot Floer homology of K. We conjecture that the homotopy type of the spectrum is an invariant of K. Our construction does not use holomorphic geometry, but rather builds on the combinatorial definition of grid homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions.

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We prove that the Khovanov spectra associated to links and tangles are functorial up to homotopy and sign.

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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.

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We prove that the module structure on Khovanov homology detects split links.

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We define stable homotopy refinements of Chen-Khovanov algebras and tangle invariants.

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We use orders of certain torsion elements in Khovanov homology to produce lower bounds on complexities of ribbon concordances between knots.

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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.

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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.

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We review the construction and context of a stable homotopy refinement of Khovanov homology.

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We define stable homotopy refinements of Khovanov's arc algebras and tangle invariants.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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This is the my PhD thesis, as submitted to the Princeton University.

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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.

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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.

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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.

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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.

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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.

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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.