arXiv research

The paper proves an inequality for symmetric polynomials under a fixed point measure.

The paper proves deep neural networks with analytic activation can approximate any function.

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek poly…

In this article, we give an elementary construction of homological invariants of links presented by braid closures. The Euler characteristic of this complex is equal to quantum polynomial invariant of link.

This is an introduction to the Volume Conjecture and its generalizations for nonexperts. The Volume Conjecture states that a certain limit of the colored Jones polynomial of a knot would give the volume of its complement. If we deform the parameter of the colored Jones polynomial we also conjecture that it would also g…

In this short note we show the existence of an epimorphism between groups of $2$-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of $2$-bridge knots by Riley polynomials.

A simplified proof of the Alexander-Conway polynomial exists.

We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.

We show how Seifert surfaces, so useful for the understanding of the Alexander polynomial Δ_L(t), can be generalized in order to study the multivariable Alexander polynomial Δ_L(t_1,...,t_μ). In particular, we give an elementary and geometric proof of the Torres formula.

The paper solves Bernstein problems for specific submanifolds in high-dimensional spaces.

The Penrose-Kauffman polynomial connects knot theory to graph coloring.

It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a lin…

Simple proof of knot genus theorem using Alexander polynomial.

New method realizes planar graphs as Reeb graphs of algebraic functions.

The paper constructs quantum invariants for knotoid diagrams.

This is a recreational paper showing that certain linked graphs cannot be separated. The proofs employ elementary covering space theory, an appeal to a theorem of Scharlemann (concerning the band sums of two unknots), and a Jones polynomial calculation.

Simplified Khovanov polynomials for bipartite links.

Paper constructs a new approach to extract Affine Index Polynomial from Sawollek Polynomial.

Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventu…

Extends double linear policy with time-varying weights and proves robust positive expectation.

We consider Conway polynomials of two-bridge links as Euler continuant polynomials. As a consequence, we obtain new and elementary proofs of classical Murasugi's 1958 alternating theorem and Hartley's 1979 trapezoidal theorem. We give a modulo 2 congruence for links, which implies the classical Murasugi's 1971 congruen…

In a previous paper, we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with <10 crossings is a simple-ribbon knot. In this paper, we give a formula for th…

We show that all twist knots, certain double twist knots and some other 2-bridge knots are minimal elements for the partial ordering on the set of prime knots. The key to these results are presentations of their character varieties using Chebyshev polynomials and a criterion for irreducibility of a polynomial of two va…

Given a virtual knot $K$, we construct a group $VG_K$ called the virtual knot group, and we use the elementary ideals of $VG_K$ to define invariants of $K$ called the virtual Alexander invariants. For instance, associated to the $k=0$ ideal is a polynomial $H_K(s,t,q)$ in three variables which we call the virtual Alexa…

The paper introduces new invariants for genus one knots and surfaces.

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…

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…

It is well known that any three-manifold can be obtained by surgery on a framed link in $S^3$. Lickorish gave an elementary proof for the existence of the three-manifold invariants of Witten using a framed link description of the manifold and the formalisation of the bracket polynomial as the Temperley-Lieb Algebra. Ka…

The paper proves curvature estimates for specific hypersurfaces in hyperbolic space.

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new struc…

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solutio…

Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some act…

This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. On the way we prove various (old and new) facts about knots. We rela…

Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of semi-algebraic proof systems that manipulate polynomial inequalities via elementary inferen…

After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5…

Study $G_2$ skein algebra elements using Kuperberg webs and threading operations.

We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which are involved …

Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an …

We define invariants of oriented surface-links by enhancing the biquandle counting invariant using \textit{biquandle modules}, algebraic structures defined in terms of biquandle actions on commutative rings analogous to Alexander biquandles. We show that bead colorings of marked graph diagrams are preserved by Yoshikaw…

Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface. We consider curvature flows in N with different curvature functions F (including the mean curvature, the gauss curvature and the second elementary symmetric polynomial) and a volume preserving term. Under suitable a…

Recently, Mullins calculated the Casson-Walker invariant of the 2-fold cyclic branched cover of an oriented link in S^3 in terms of its Jones polynomial and its signature, under the assumption that the 2-fold branched cover is a rational homology 3-sphere. Using elementary principles, we provide a similar calculation f…

The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated topological entropy is the logarithmic growth rate of the number of torsion elem…

In this paper we study the problem of finding a conformal metric with the property that the k-th elementary symmetric polynomial of the eigenvalues of its Weyl-Schouten tensor is constant. A new conformal invariant involving maximal volumes is defined, and this invariant is then used in several cases to prove existence…

We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers is that we deal here with an elementary, interesting, new mathematics, an…

A second part of detailed elementary introduction into Khovanov homologies. This part is devoted to reduced Jones superpolynomials. The story is still about a hypercube of resolutions of a link diagram. Each resolution is a collection of non-intersecting cycles, and one associates a 2-dimensional vector space with each…

We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically SL(n,Z) and Sp(2n, Z) to show that a ``ran…

The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell. The game on a graph $G$ can be described as follows: let $f_n$ be a sequence of positive integers; an initial fire starts at a finite set of vertices; at each (integer) time $n\geq 1$, $f_n$ vertices which are not on fire b…

The study examines differential singularities in 3D Nijenhuis operators.