arXiv research

Proves divisibility relations for symplectic curve polynomials.

This paper investigates the equivalence between Yamada polynomial and Jones polynomial of associated links for brunnian θ-curves.

Determines the crossing number of polynomial curve systems on surfaces.

New method measures entanglement of open and closed curves.

The normalized Yamada polynomial is a polynomial invariant in variable A for theta-curves. In this work, we show that the coefficients of the power series obtained from this polynomial by the substitution A=e^x=1+x+x^2/2+x^3/6+... are finite-type invariants for theta-curves although the coefficients of original polynom…

Novel Jones polynomial for open curves in 3D space.

Polynomial decay of correlations shown for curved surfaces.

New knot models analyze local entanglement for robust curve analysis.

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…

Classifies special homogeneous curves with polynomial equations.

Analytic functions on specific domains are characterized by their smoothness and composites with polynomial curves.

We study a family of polynomials in two variables having moduli up to bilipschitz equivalence: two distinct polynomials of this family are not bilipschitz equivalent. However any level curve of the first polynomial is bilipschitz equivalent to a level curve of the second.

The study finds resonance points in polarised curves with polynomial conserved quantities.

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the…

Polynomially inscribe 6 concyclic points into any smooth curve.

In this paper we define a new type of 2-degenerate Cartan curves in Minkowski spacetime $R_{1}^{4}$. We prove that this type of curves contain only the polynomial functions as its components whose third derivative vanish completely. No curve with acceleration zero in $R_{1}^{4}$ is a 2-degenerate Cartan curve, therefor…

Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Con…

For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2,C)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2,C)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.…

Algorithm classifies surface homeomorphisms with polynomial time complexity.

We introduce a Kauffman-Jones type polynomial $\mathcal{L}_γ(A)$ for a curve $γ$ on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial $\mathcal{L}_γ(A)$ is a Laurent polynomial in one variable $A$ and is an invariant of the homotopy class of $γ$. As an application, we obtain an est…

Study on frequencies of non-simple curves in surfaces of large genus.

Defines Vassiliev complexity measures for open and closed curves in 3D space.

Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either $\pm1$ or $0$ and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's p…

We give two formulae which express the Alexander polynomial $Δ^C$ of several variables of a plane curve singularity $C$ in terms of the ring ${\cal O}_{C}$ of germs of analytic functions on the curve. One of them expresses $Δ^C$ in terms of dimensions of some factorspaces corresponding to a (multi-indexed) filtration o…

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…

Polynomial bound on tightening curves on surfaces without increasing crossings.

In this paper we address the following questions: (i) Let $C\subset \mathbb C^2$ be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is $C$ contained in an algebraic curve? (ii) What can be said of a polynomial vector field which has a finitely curved transcendent orbit? We give a positi…

We analyze relations between BPS degeneracies related to Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the correspond…

New method detects projective equivalences and symmetries in rational 3D curves.

A polynomial curve of degree 5, $α$, is a helix, if and only if both $||α^'||$ and $||α^'\wedge α^{''}||$ are polynomial functions.

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

Classifies special quartic curves up to equivalence.

By analyzing the affine Taylor expansion of a non-degenerate plane curve, we obtain characterizations of classes of such curves via curvature properties of the gravity curve. The proof is based on an analysis of the degree parity and leading coefficients of polynomials occurring in the expansion.

We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [KP10] and [BKP14]. We also give a sharp…

Topological recursion recovers a specific partition function for colored knots.

Study Kähler geometry on vector bundles over elliptic curves.

We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brezín and Hikami. We express this recursion in special times in which all terms $W_1^{(g)}$ of the genus expansion of the one-loop mean are polynomials. We find a generalization of th…

New polynomials defined for virtual knots, calculated up to crossing 4.

We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We recover the classical torsionness and divisibility results, which say that, unde…

Explicit polynomial bound found for subgroup Dehn function.

We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular, we use it to give a complete classification of biharmonic curves into real quadr…

The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.

We study the twisted Alexander polynomial from the viewpoint of the SL(2,C)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL(2,C)-representations are all monic. In this paper, we show that the con…

We use topological methods to study various semicontinuity properties of spectra of singular points of plane algebraic curves and of polynomials in two variables at infinity. Using Seifert forms and the Tristram--Levine signatures of links, we reprove (in a slightly weaker version) a result obtained by Steenbrink and V…

A topological invariant of a polynomial map $p:X\to B$ from a complex surface containing a curve $C\subset X$ to a one-dimensional base is given by a rational second homology class in the compactification of the moduli space of genus $g$ curves with $n$ labeled points $\modmgn$. Here the generic fibre of $p$ has genus …

In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that thi…

We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top degree coefficients tautological intersection numbers on the compactified moduli spac…

The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted $q$ or $\hbar$; this quan…