arXiv research

Study on polynomial growth functions and forms on gradient Ricci solitons.

As is well-known, the Witten deformation of the De Rham complex computes the De Rham cohomology. In this paper we study the Witten deformation on a noncompact manifold and restrict it to differential forms which behave polynomially near infinity. Such polynomial differential forms naturally appear on manifolds with a c…

Study shows knots with similar Blanchfield forms can be homotopy ribbon concordant.

Clock theorem extended to knotoids and linkoids.

We define and calculate the HOMFLY polynomial for a specific type of quiver.

New methods compute Alexander polynomials for complex knots.

A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor $π^+$ corresponding to $\om^+$ is po…

We show how the signed evaluations of link polynomials can be used to calculate unknotting numbers. We use the Jones-Rong value of the Brandt-Lickorish-Millett-Ho polynomial Q to calculate the unknotting numbers of 8_{16}, 9_{49} and 6 further new entries in Kawauchi's tables. Another method is developed by applying an…

We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingred…

This article discuss a class of tractable model in the form of polynomial type.

Simpler equations derived for knot polynomials coefficients, forming a ring.

The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees th…

Origami structures are enumerated and shown to be quantum modular.

The paper proves positivity of characteristic forms for certain vector bundles.

Paper corrects a proof about biharmonic hypersurfaces with three distinct curvatures.

Suppose the knot group G(K) of a knot K has a non-abelian representation ρon A_4 \subset GL(4,Z). We conjecture that the twisted Alexander polynomial of K associated to ρis of the form: Δ_K(t)/(1-t) φ(t^3), where Δ_K (t) is the Alexander polynomial of K and φ(t^3) is an integer polynomial in t^3. We prove the conjectur…

Derives adjoint polynomials of torus knots in explicit form.

We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.

Novikov initiated the study of the algebraic properties of quadratic forms over polynomial extensions by a far-reaching analogue of the Pontrjagin-Thom transversality construction of a Seifert surface of a knot and the infinite cyclic cover of the knot exterior. In this paper the analogy is applied to explain the relat…

The Alexander polynomials Δ_{n,3}(t) and Δ_{n,4}(t) are presented as a sum of the Alexander polynomials Δ_{k,2}(t). These polynomials are also expressed in the form of a sum of Chebyshev polynomials of the second kind. These expansions allow one to introduce the "coordinates" in corresponding bases, which are proposed …

New knot polynomials yield simple results modulo primes.

Proves a formula for push-forward of polynomial Chern forms in universal vector bundles.

Study Alexander polynomials of modular knots, revealing finite and infinite coefficient properties.

Geometrically describes the linear and quadratic forms for rational links.

Closed-form polynomial approximations replace MLPs in transformers, enabling new interpretability methods.

Study on spatial graphs and their constituent knots, linking polynomial invariants.

Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.

Study proves Alexander polynomials of certain 4-braid knots satisfy a conjecture and gives formulas for log-concave sequences.

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…

In the present paper we discuss the cabling procedure for the colored HOMFLY polynomial. We describe how it can be used and how one can find all the quantities such as projectors and $\mathcal{R}$-matrices, which are needed in this procedure. The constructed matrix forms of the projectors and the fundamental $\mathcal{…

Jones polynomial for twisted torus knots is trivial if and only if the knot is trivial.

The paper studies SDP feasibility and sos ranks for specific polynomials.

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions are constructed in all dimensions and solutions are classified in d…

Using the Fourier expansion of Markov traces for Ariki-Koike algebras over $\mathbb{Q}(q,u_{1},...,u_{e})$, we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite simple …

We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: $\mathcal{A}^\mathcal{K}_R(q)=\mathcal{A}^\mathcal{K}_{[1]}(q^{\vert R\vert})$ for all 1-hook Young diagrams $R$. Via the Kontsevich construction, it is reformulated …

We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system.

We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special Kähler geometry and the ring of almost-holomorphic modular forms defined on modular curves.

The paper characterizes biharmonic maps between spheres using polynomial functions.

Algorithm calculates Jones polynomial from Goeritz matrix.

Researchers compute and predict knot volumes using colored Jones polynomials.

Study geometric structures and their interactions under different metrics.

Constructs finite element spaces for $(p,q)$-forms, excluding one subspace.

Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.

We implement a differential-geometric approach to normal forms for contracting measurable cocycles to $\mbox{Diff}^q({\bf R}^n, {\bf 0})$, $q \geq 2$. We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via $C^q$ changes of coordinates. These are interpreted as nonstationary inv…

A formula for triangle area in Deep Sets form.

We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, and give their ex…

Polynomial algorithm for multiplication on one-hole torus skein algebra.

Novel link classification connects quadratic forms and knot theory.