arXiv research

We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensor…

For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and Kálmán constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$-th Catalan number. We show that these exact Lagrangian fillings are pairwise non-isotopic through exact Lagrangian isotopy. To do that, we com…

Study shows codimension 2 foliations on simply-connected manifolds are smooth.

In this article we study the Heegaard Floer link homology of $(n, n)$-torus links. The Alexander multigradings which support non-trivial homology form a string of $n-1$ unit hypercubes in $\mathbb{R}^{n}$, and we compute the ranks and gradings of the homology in nearly all Alexander gradings. We also conjecture a compl…

Optimal bounds found for torus curvatures in high dimensions.

The paper characterizes gaps in minimal foliations on tori using energy criteria.

In this paper, we study the Khovanov homology of cable links. We first estimate the maximal homological degree term of the Khovanov homology of the ($2k+1$, $(2k+1)n$)-torus link and give a lower bound of its homological thickness. Specifically, we show that the homological thickness of the ($2k+1$, $(2k+1)n$)-torus li…

We complete the classification of compact connected contact toric manifolds initiated by Banyaga and Molino and by Galicki and Boyer. As an application we prove the conjectures of Toth and Zelditch on toric integrable systems on the n-torus and the 2-sphere.

Symmetric function $L_{M,N}$ lifts torus link homology.

Proves a cube cannot have positive scalar curvature with specific conditions.

We discuss the construction of toric Kaehler metrics on symplectic 2n-manifolds with a hamiltonian n-torus action and present a simple derivation of the Guillemin formula for a distinguished Kaehler metric on any such manifold. The results also apply to orbifolds.

Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, …

Extended Thistlethwaite's result on Jones polynomials of quasi-alternating links.

The paper connects three integer sequences to a knot polynomial.

The main results of this paper are an asymptotic expansion in powers of $\hbar$ for the spectral measure $μ_\hbar$ of a semi-classical Toeplitz operator, $Q_\hbar$, and an equivariant version of this result when $Q_\hbar$ admits an $n$-torus as a symmetry group. In addition we discuss some inverse spectral consequences…

We introduce a new method for computing triply graded link homology, which is particularly well-adapted to torus links. Our main application is to the (n,n)-torus links, for which we give an exact answer for all n. In several cases, our computations verify conjectures of Gorsky et al relating homology of torus links wi…

New findings on minimal isometric immersions of flat n-tori into spheres.

This article announces joint work with Frank Connolly and Jim Davis. We generalize our classification of pseudo-free involutions on the n-torus, by studying the action of the associated infinite group with torsion in the universal cover. Included is a non-Riemannian example obtained from the restriction of the action o…

We conjecture an expression for the dimensions of the Khovanov-Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture relating the HO…

The n-dimensional torus is uniquely characterized by specific harmonic forms.

The paper studies traceless SU(2) representations and instanton gradings for specific knot types.

The m,n Turk's Head Knot, THK(m,n), is an "alternating (m,n) torus knot." We prove the Harary-Kauffman conjecture for all THK(m,n) except for the case where m \geq 5 is odd and n \geq 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that non…

An L-space link is a link in $S^3$ on which all sufficiently large integral surgeries are L-spaces. We prove that for m, n relatively prime, the r-component cable link $K_{rm,rn}$ is an L-space link if and only if K is an L-space knot and $n/m \geq 2g(K)-1$. We also compute HFL-minus and HFL-hat of an L-space cable lin…

We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield and the third author receive an explicit algebra…

We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space R^{4d} by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an n-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration o…

Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsions. When the underlying algebra is $\mathbb{Z}[x]/(x^2)$, we determine precisely those graphs whose cohomology contains torsion. For a la…

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some universal constant depending on $n$. He obtained similar estimates for maps with va…

Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde M$ satisfies the inequality $\dim_{mc}\tilde M\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duali…

Let M be a Kaehler manifold with a free, holomorphic and Hamiltonian action of the standard n-torus T. We give a simple, explicit and canonical formula for the Kaehler potential on the Kaehler reduction of M. As a consequence we can derive improvements of several classical results known for more general Hamiltonian red…

The paper identifies all link projections with isolate-region number one.

Classifies actions of SL(n,R) and SL(n,Z) on closed n-manifolds.

We prove there is only one involution (up to conjugacy) on the n-torus which acts as $-\mathrm{Id}$ on the first homology group when $n$ is of the form $4k$, is of the form $4k+1$, or is less than $4$. In all other cases we prove there are infinitely many such involutions up to conjugacy, but each of them has exactly $…

The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We …

Jones polynomial gaps removed for quasi-alternating links.

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorph…

The study proves manifold properties related to positive scalar curvature.

A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2π\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map $φ:M\to\R^n$, a …

We prove that if the lens space $L(n, 1)$ is obtained by a surgery along a knot in the lens space $L(3,1)$ that is distance one from the meridional slope, then $n$ is in $\{-6, \pm 1, \pm 2, 3, 4, 7\}$. This result yields a classification of the coherent and non-coherent band surgeries from the trefoil to $T(2, n)$ tor…

Let $D(p,q)$ be the usual knot diagram of the $(p,q)$-torus knot, that is, $D(p,q)$ is the closure of the $p$-braid $(σ_1^{-1} σ_2^{-1}... σ_{p-1}^{-1})^q$. As is well-known, $D(p,q)$ and $D(q,p)$ represent the same knot. It is shown that $D(n+1,n)$ can be deformed to $D(n,n+1)$ by a sequence of $\{(n-1)n(2n-1)/6 \} + …

Obstructs complete metrics with positive scalar curvature on non-compact manifolds.

For any given integer $r \geq 1$ and a quasitoric braid $β_r=(σ_r^{-ε} σ_{r-1}^ε...$ $σ_{1}^{(-1)^{r}ε})^3$ with $ε=\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2(\hatβ_r)}(v,z)$ of the doubled link $W_2(\hatβ_r)$ of the closure $\hatβ_r$ is equal to $6r-1$. As an application, we gi…

The k-th Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, V_k(A), of the complex algebraic n-torus. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of V_k…

A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including $(2, n)$ torus knots, $2$-bridge knots, algebraic alter…

We reconsider the su(3) link homology theory defined by Khovanov in math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of (complexes of) `cobordisms with seams' (actually, a `canopo…

Explicit matrix presentations of Blanchfield pairings and twisted pairings for torus knots.

Constructs pseudo-Anosov mappings related to toral automorphisms.

A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,α)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $α:\mathbb{T}^n\rightarrow\Aut(A)$, which induces a (smooth) continuous action of $\mathbb{T}^n$ on $A$. In this…

We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a $(2,n)$-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connectio…