arXiv research

Quantum polynomials are derived from a specific tribracket structure.

Study the relationship between canonical polynomials and elliptic sequences for elliptic singularities.

This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.

Proves freely 2-periodic knots have two canonical components in their character variety.

Computes minimal polynomials for generalized Heisenberg groups.

We give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of the slope of the given rational link.

We compute the A-polynomial 2-tuple of twisted Whitehead links. As applications, we determine canonical components of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds.

Polynomial-time algorithm finds short non-orientable loops intersecting graph edges up to 30 times.

Researchers extend Alexander polynomial to knotoids and linkoids.

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for…

We describe a polynomial-time algorithm to compute a (tight) geodesic between two curves in the curve graph. As well as enabling us to compute the distance between a pair of curves, this has several applications to mapping classes. For example, we can use these geodesics to compute the asymptotic translation length, Ni…

A string link S can be closed in a canonical way to produce an ordinary closed link L. We also consider a twisted closing which produces a knot K. We give a formula for the Conway polynomial of L as a product of the Conway polynomial of K times a power series whose coefficients are given as explicit functions of the Mi…

The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that th…

We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an immersion (e.g. regular diagrams of a classical knot in the complement of its meridi…

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomia…

We prove that the mapping torus group $\FN \rtimes_α \Z$ of any automorphism $α$ of a free group $\FN$ of finite rank $n \geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(α)$ of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that …

For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geome…

New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.

Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $θ\colon \mathbb{C}^n\to \mathbb{C}^n$ which outputs the critical values of the canonical polynomial constructed from the…

We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links b…

We give a simple algorithm that determines whether a given post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, the algorithm produces the Hubbard tree; otherwise, the algorithm produces the canonical obstruction. Our approach is rooted in geometric group theory, using iterati…

We describe a procedure for creating infinite families of knots, each having the maximum degree of their HOMFLY polynomial strictly less than twice their canonical genus. These families build upon examples first found by Stoimenow.

The paper calculates the Hilbert polynomials for configuration spaces over graphs with a short circumference.

Paper introduces equations to distinguish knots without using knot invariants.

We introduce canonical measures on a locally finite simplicial complex $K$ and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the $d^{th}$ barycentric subdivision $Sd^d(K)$ of $K$, $d\gg0$. It is a…

New Poisson structures on hypersurface algebroids discovered.

We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a certain rate towards the iden…

We study the change of moduli spaces of Gieseker-semistable torsion free rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the surfaces are rational with an effective anti-canonical divisor, the moduli spaces are linked by a series of flips (blowups and blowdowns). Using these results, we comput…

The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces…

Khovanov homology is a recently introduced invariant of oriented links in $\mathbb{R}^3$. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov homology is a version of the Jones polynomial for links. In this paper we study torsion of the Khovanov homology. Based on ou…

A polynomial invariant for veering triangulations helps in understanding 3-manifold fibers.

We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several exa…

New geometric object for polynomials simplifies complex data.

Paper identifies reductive MDPs, solving them in polynomial time.

The paper constructs triangulations for double twist knots using geometric methods.

We investigate the structure of a variety of new Moishezon twistor spaces, by utilizing the pluri-half-anti-canonical map from the twistor spaces. Each of these twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of…

Study shows polynomial-width neural networks can closely approximate infinite-width networks in polynomial time.

We study certain polynomial trace identities in the group $SL(2,\IC)$ and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in §4 and then in §5 we give a new proof for Gehring and Martin's polynomial trace identities for good words, and extend that result …

On a Fano manifold, we prove that the Kahler-Ricci flow starting from a Kahler metric in the anti-canonical class which is sufficiently close to a Kahler-Einstein metric must converge in a polynomial rate to a Kahler-Einstein metric. The convergence can not happen in general if we study the flow on the level of Kahler …

Continuity of polynomial roots shown for varying coefficients.

Study minimal timelike surfaces in 3D Lorentz-Minkowski space using holomorphic functions.

We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {Š}evera. We study the relationship of odd symplectic geometry with classical objects. We show that the Berezinian of a canonical transformatio…

A virtual knot that has a homologically trivial representative $\mathscr{K}$ in a thickened surface $Σ\times [0,1]$ is said to be an almost classical (AC) knot. $\mathscr{K}$ then bounds a Seifert surface $F\subset Σ\times [0,1]$. Seifert surfaces of AC knots are useful for computing concordance invariants and slice ob…

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…

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

A 2-step nilpotent Lie algebra n is called nonsingular if ad(X): n --> [n,n] is onto for any X not in [n,n]. We explore nonsingular algebras in several directions, including the classification problem (isomorphism invariants), the existence of canonical inner products (nilsolitons) and their automorphism groups (maxima…

New knot polynomials distinguish knot orientations without using knot groups.

TensorSketch is an oblivious linear sketch introduced in Pagh'13 and later used in Pham, Pagh'13 in the context of SVMs for polynomial kernels. It was shown in Avron, Nguyen, Woodruff'14 that TensorSketch provides a subspace embedding, and therefore can be used for canonical correlation analysis, low rank approximation…