arXiv research

Conditions for integer signatures of high-dimensional knots.

Turning the skein relation for HOMFLY into a Fibonacci recurrence, we prove that there are only three rational specializations of HOMFLY polynomial: Alexander-Conway, Jones, and a new one. Using the recurrence relation, we find general and relative expansion formulae and rational generating functions for Alexander-Conw…

The study proves conjecture for specific Artin groups.

Cyclotomic polynomials help classify mapping classes on surfaces.

New invariant for square-free integers derived from kei theory.

We give a complete classification of homomorphisms from the braid group on $n$ strands to the braid group on $2n$ strands when $n$ is at least 5. We also classify endomorphisms of the braid group on 4 strands, as well as homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on …

Let $n$ be a positive integer, and let $\ell>1$ be square-free odd. We classify the set of equivariant homeomorphism classes of free $C_\ell$-actions on the product $S^1 \times S^n$ of spheres, up to indeterminacy bounded in $\ell$. The description is expressed in terms of number theory. The techniques are various appl…

We study a subset of square free positive braids and we give a few algebraic characterizations of them and one geometric characterization: the set of positive braids whose closures are unlinks. We describe canonical forms of these braids and of their conjugacy classes.

Band surgery is an operation relating pairs of knots or links in the three-sphere. We prove that if two quasi-alternating knots $K$ and $K'$ of the same square-free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a …

New lattices in higher rank contain a fixed 3-manifold group with increasing systole.

In the 1970s, Birman-Craggs-Johnson used Rochlin's invariant for homology 3-spheres to construct a remarkable surjective homomorphism sigma:I_{g,1}->B_3, where I_{g,1} is the Torelli group and B_3 is a certain F_2-vector space of Boolean (square-free) polynomials. By pulling back cohomology classes and evaluating them …

Let $H$ be a hyperexponential function in $n$ variables $x=(x_1,\dots,x_n)$ with coefficients in a field $\mathbb{K}$, $[\mathbb{K}:\mathbb{Q}] <\infty$, and $ω$ a rational differential $1$-form. Assume that $Hω$ is closed and $H$ transcendental. We prove using Schanuel conjecture that there exist a univariate function…

Let $d$ be a positive square-free integer $\equiv 3 \pmod{4}$ such that there is no invariant of the ideal class group $\mathbb{Q}\lbrack \sqrt{-d}\rbrack$ which is divisible by $4$. We prove an asymptotic formula for the number of immersed totally geodesic surfaces in $Γ_{-d}\backslash \mathbb{H}^3$ having area less t…

In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f…

The study counts Salem numbers linked to arithmetic hyperbolic orbifolds.

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $Σ_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $Σ_g$. For …

The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology sphere…

Given knots K and J, one can ask whether a single smoothing of a crossing in a diagram for K can convert it into a diagram for J. As an interesting example, Zekovic discovered that the torus knot T(2,5) can be converted into T(2,-5) with a single smoothing. On the other hand, Moore and Vasquez have shown that among tor…

Let $Γ<\mathrm{SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $Γ(q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of $L^2(Γ(q) \backslash \mathrm{SL}_2(\mathbb{R}))$ with a {\it uniform} exponential error term…

Algorithm estimates covariance from noisy data efficiently.

New $F$-polynomial distinguishes knotoid diagrams not previously possible.

The paper studies polynomials and ideals from colored Jones polynomials for links.

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

This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.

Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.

The paper defines and classifies Cappell-Shaneson polynomials.

Developed algorithms to compute three polynomial invariants of veering triangulations.

Study links weaving knots with polynomial coefficients and lattice numbers.

Study revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.

Paper connects AJ conjecture and colored Jones polynomial potential function.

Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a numb…

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

The taut polynomial equals a twisted Alexander polynomial.

Quantum polynomials are derived from a specific tribracket structure.

We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomial…

New polynomials detect non-rotatable knotoid shapes.

Innovates polynomial invariant for tribrackets.

Researchers extend Alexander polynomial to knotoids and linkoids.

Unified ADO and colored Jones polynomials for knots.

Generalized quandle polynomial used for stuquandles, stuck links, and RNA folding.

Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.

We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that $ns^at^a$-quandles are not classified by their generalized quandle polynomials. We use subrack polynomials to define enhanced rack counting invariants, gen…

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author. The relation between this…

Paper computes Alexander polynomials for arborescent links.

Jones polynomials derived from K-theory of a cluster algebra.

We study relationships between the colored Jones polynomial and the A-polynomial of a knot. We establish for a large class of 2-bridge knots the AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial. Along the way we also calculate the Kauffman bracket skein module of all 2-brid…

The interior polynomial is an invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to a par…

New skein theory for Links-Gould polynomial simplifies link evaluations.