arXiv research

New knot polynomials yield simple results modulo primes.

The paper defines and classifies Cappell-Shaneson polynomials.

Upper bound on Jones polynomials density modulo primes.

We discuss techniques for analysing the structure of the group obtained by reducing the image of the Burau representation of the braid group modulo a prime. The main tools are a certain sesquilinear form first introduced by Squier and consideration of the action of the group on a Euclidean building.

The Burau representation of 3-strand braid group modulo p is determined and shown to be faithful for small p.

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant. Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2…

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knots of type (2, p), modulo p. This expression is t + 2 l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations…

We analyze two braid group representations and their reductions modulo p.

It is known that the first two-variable Links--Gould quantum link invariant $LG\equiv LG^{2,1}$ is more powerful than the HOMFLYPT and Kauffman polynomials, in that it distinguishes all prime knots (including reflections) of up to 10 crossings. Here we report investigations which greatly expand the set of evaluations o…

We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their abelianization (or equivalently, their integer homology class. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev pol…

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of Fox colors is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove …

We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a rando…

We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of links in L(p,q) as a function of the level r-2. They are given by 1 over the square root of r times one of p Laurent polynomials evaluated at e to the 2 pi i divided by 4pr. The congruence class of r modulo p determines which polynomial is appli…

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $λ$-symmetries (C. Muriel and J. L. Romero, \emph{IMA J. Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE $\mathcal{Y}$ is used here to recover $λ$-symmetries of $\mathcal{Y}$ as nonlocal symmetries. In …

This article is about applications of linear algebra to knot theory. For example, for odd prime p, there is a rule (given in the article) for coloring the arcs of a knot or link diagram from the residues mod p. This is a knot invariant in the sense that if a diagram of the knot under study admits such a coloring, then …

In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6 for p \geq 17. We elaborate on equivalence classes of sets of distinct colors (o…

The paper determines the maximal order of translation groups in abelian differentials for various genera.

We show that every periodic virtual knot can be realized as the closure of a periodic virtual braid and use this to study the Alexander invariants of periodic virtual knots. If $K$ is a $q$-periodic and almost classical knot, we show that its quotient knot $K_*$ is also almost classical, and in the case $q=p^r$ is a pr…

In this paper, a vanishing theorem is stated and proved. If a 4-manifold $M$ admits a smooth action by a cyclic group $\mathbb{Z}_r$, then given an $\mathbb{Z}_r$-equivariant $Spin^c$-structure $\mathcal{C}$ on $M$, the Seiberg-Witten invariant $SW\mathcal{(C)}$ is zero modulo $r$ under some slight assumptions. Here $r…

Cobordism groups of cooriented fold maps of codimension 1 are computed completely. Namely their odd torsion part coincides with that of the stable homotopy group of spheres in the same dimension, while the 2-primary part is the kernel of the Kahn-Priddy map. (The Kahn-Priddy map is an epimorhism of the stable homotopy …

In a strengthening of the G-Signature Theorem of Atiyah and Singer, we compute, at least in principle (modulo certain torsion of exponent dividing a power of the order of G), the class in equivariant K-homology of the signature operator on a G-manifold, localized at a prime idea of R(G), in terms of the classes in non-…

Paper explores relationships between triple chords and a specific homotopy relation in knot theory.

Study invariants of $\mathbb{Z}/p$-homology 3-spheres from abelianization of mapping class groups.

A natural p-classes generalization of the eXclusive OR problem, the subtraction modulo p, where p is prime, is presented and solved using a single fully connected hidden layer with p-neurons. Although the problem is very simple, the landscape is intricate and challenging and represents an interesting benchmark for grad…

The classical prime geodesic theorem (PGT) gives an asymptotic formula (as $x$ tends to infinity) for the number of closed geodesics with length at most $x$ on a hyperbolic manifold $M$. Closed geodesics correspond to conjugacy classes of $π_1(M)=Γ$ where $Γ$ is a lattice in $G=SO(n,1)$. The theorem can be rephrased in…

Given an oriented rational homology 3-sphere M, it is known how to associate to any Spin^c-structure σon M two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo 1 of the Reidemeister-Turaev torsion of (M,σ), while the other one can be defined using the intersectio…

Minkowski's second theorem can be stated as an inequality for $n$-dimensional flat Finsler tori relating the volume and the minimal product of the lengths of closed geodesics which form a homology basis. In this paper we show how this fundamental result can be promoted to a principle holding for a larger class of Finsl…

We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There i…

Przytycki and Sokolov proved that a three-manifold admits a semi-free action of the finite cyclic group of order $p$ with a circle as the set of fixed points if and only if $M$ is obtained from the three-sphere by surgery along a strongly $p-$periodic link $L$. Moreover, if the quotient three-manifold is an integral ho…

In this paper, we study Mabuchi metrics on Fano manifolds. We prove that Mabuchi metrics exist if the modified Ding functional is proper modulo a reductive subgroup of its automorphism group. On the other hand, the inverse that Mabuchi metrics implies the properness is obtained by using Darvas-Rubinstein's properness p…

Let $Γ=\langle a,b | a b^{p} a^{-1} = b^{q}\rangle$ be a Baumslag--Solitar group and $G$ be a complex reductive algebraic group with maximal compact subgroup $K<G$. We show that, when $p$ and $q$ are relatively prime with distinct absolute values, there is a strong deformation retraction retraction of $Hom(Γ,G)$ onto $…

Hasse principle applied to area-minimizing submanifolds across different homology types.

The paper connects ADO polynomials to Vassiliev invariants for knots.

The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find …

In this paper we extend recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of constant scalar Kähler metric on a compact Kähler manifold to Calabi's extremal metric. Our argument follows \cite{CC3} and there are no new a prior estimates needed, but rather there are necessary modifications adapted to th…

The paper studies dynamical properties in semigroups modulo ideals.

I. Hambleton, A. Korzeniewski and A. Ranicki proved that the signature of a fibre bundle of closed, connected, compatibly oriented PL manifolds is always multiplicative modulo 4. In this paper, we consider the Hirzebruch $χ_y$-genera for odd integers $y$ for a smooth fiber bundle such that the base, fibre, and total sp…

Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p, t^2 + kappa t + 1), where p is a prime number and t^2 + kappa t + 1 is irreduci…

Study on geodesics on random hyperbolic surfaces, showing variance asymptotic to X log X.

Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-for…

We construct non-trivial elements of order 2 in the homotopy groups $π_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Mor…

We compute the groups $H^*(\mathrm{Aut}(F_n); M)$ and $H^*(\mathrm{Out}(F_n); M)$ in a stable range, where $M$ is obtained by applying a Schur functor to $H_\mathbb{Q}$ or $H^*_\mathbb{Q}$, respectively the first rational homology and cohomology of $F_n$. For reasons which are not conceptually clear, taking coefficient…

We study the dependence of the eta invariant $η_D$ on the spin structure, where $D$ is a twisted Dirac operator on a (4k+3)-dimensional spin manifold. The difference between the eta invariants for two spin structures related by a cohomolgy class which is the reduction of a $H^1(M,\Za)$-class is shown to be a half integ…

Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.

Modulo-SK outperforms Deepcode in error probability and feedback rounds.

Proves prime theta-curves for knots on minimal genus surfaces.

We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in terms of the structure of the braid groups. We also prove some results about kno…

Counterexamples show Salter's question on Burau image is negative for n=4.