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Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we prove that reduced Khovanov homology (over $\mathbb{Q}$) detects the figure-eight knot.

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms., Comment: 28 pages, 16 figures, 1 table

We prove that a positive definite smooth four-manifold with $b_2^+ \geq 2$ and having either no 1-handles or no 3-handles cannot admit a symplectic structure.

The cosmetic crossing conjecture (also known as the “nugatory crossing conjecture”) asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.

We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to$S^{2}$but do not admit a spine (that is, a piecewise linear embedding of$S^{2}$that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer$d$invariants.

We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integer homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f : Y ′ → Y f:Y’ \to Y between Seifert homology spheres yields the inequality | deg ⁡ f | r a n k H F r e d ( Y ) ≤ r a n k H F r e d ( Y ′ ) |\deg f|\mathrm {rank} HF_{\mathrm {red}}(Y) \leq \mathrm {rank} HF_{\mathrm {red}}(Y’) . These inequalities are also applied in conjunction with an algorithm of Némethi to give a method to solve the botany problem for the Heegaard Floer homology of these manifolds.

We correct an error in the statement and proof of Theorem 1.4 of our paper in Acta Mathematica Vietnamica (2014) 39(4), 599-635.

Quasi-alternating links of determinant 1, 2, 3 and 5 were previously classified by Greene and Teragaito, who showed that the only such links are two-bridge. In this paper, we extend this result by showing that all quasi-alternating links of determinant at most 7 are connected sums of two-bridge links, which is optimal since there are quasi-alternating links not of this form for all larger determinants. We achieve this by studying their branched double covers and characterising distance-one surgeries between lens spaces of small order, leading to a classification of formal L-spaces with order at most 7.

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g must have slope 2g-1, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston-Bennequin numbers of cables., Comment: 33 pages

A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links., Comment: 24 pages, 11 figures, 1 table

Ozsvath and Szabo construct a spectral sequence with E_2 term \Lambda^*(H^1(Y;Z))\otimes Z[U,U^{-1}] converging to HF^\infty(Y,s) for a torsion Spin^c structure s. They conjecture that the differentials are completely determined by the integral triple cup product form via a proposed formula. In this paper, we prove that HF^\infty(Y,s) is in fact determined by the integral cohomology ring when s is torsion. Furthermore, for torsion Spin^c structures, we give a complete calculation of HF^\infty with mod 2 coefficients when b_1 is 3 or 4., Comment: 18 pages, 5 figures

An acyclic mapping from an $n$ element set into itself is a mapping $\phi$ such that if $\phi^k(x) = x$ for some $k$ and $x$, then $\phi(x) = x$. Equivalently, $\phi^\ell = \phi^{\ell+1} = ...$ for $\ell$ sufficiently large. We investigate the behavior as $n \to \infty$ of a Markov chain on the collection of such mappings. At each step of the chain, a point in the $n$ element set is chosen uniformly at random and the current mapping is modified by replacing the current image of that point by a new one chosen independently and uniformly at random, conditional on the resulting mapping being again acyclic. We can represent an acyclic mapping as a directed graph (such a graph will be a collection of rooted trees) and think of these directed graphs as metric spaces with some extra structure. Heuristic calculations indicate that the metric space valued process associated with the Markov chain should, after an appropriate time and ``space'' rescaling, converge as $n \to \infty$ to a real tree ($\R$-tree) valued Markov process that is reversible with respect to a measure induced naturally by the standard reflected Brownian bridge. The limit process, which we construct using Dirichlet form methods, is a Hunt process with respect to a suitable Gromov-Hausdorff-like metric. This process is similar to one that appears in earlier work by Evans and Winter as the limit of chains involving the subtree prune and regraft tree (SPR) rearrangements from phylogenetics., Comment: 26 pages, 4 figures

We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $L^2$ the expected value is the orthogonal projection onto the constants. Previous work has shown that the local field version of $L^\infty$ is the appropriate counterpart of $L^2$, and so the expected value of a local field-valued random variable is defined to be its ``projection'' in $L^\infty$ onto the constants. Unlike the real case, the resulting projection is not typically a single constant, but rather a ball in the metric on the local field. However, many properties of this expectation operation and the corresponding conditional expectation mirror those familiar from the real-valued case; for example, conditional expectation is, in a suitable sense, a contraction on $L^\infty$ and the tower property holds. We also define the corresponding notion of martingale, show that several standard examples of martingales (for example, sums or products of suitable independent random variables or ``harmonic'' functions composed with Markov chains) have local field analogues, and obtain versions of the optional sampling and martingale convergence theorems., Comment: 19 pages