arXiv research

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

The main purpose of this note is the study of the total space of a holomorphic Lie algebroid $E$. The paper is structured in three parts. In the first section we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The se…

Listed 19,513 prime knots with arc index 12-16.

Classifies doodles into prime and super prime types, describing them with doodle codes.

Homogeneous braids are visually prime, solving a Cromwell question.

Kirby and Lickorish showed that every knot in the 3-sphere is concordant to a prime knot, equivalently, every concordance class contains a prime knot. We prove here that their result can be strengthened: Every knot in the 3-sphere is invertibly concordant to a prime knot. A consequence is that every double concordance …

This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossing…

This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (T …

Computed the 4-genus for all 12-crossing prime knots.

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 establish an algorithm which computes formulae for the CR GJMS operators, the $P^\prime$-operator, and the $Q^\prime$-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the $P^\prime$-operator…

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish in [3]; Lickorish proved [5] that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produc…

Minimal grid diagrams for 12-crossing prime knots identified.

The paper generalizes CR invariants using renormalized characteristic forms.

We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime surface-unknot and surface-unlink diagrams with prime base surface components up to …

New method proves topological Tverberg problem for all q, not just primes.

Upper bound on Jones polynomials density modulo primes.

Study of origamis' singularities for groups of prime-power order.

New invariant for prime alternating knots from error-correcting codes

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theor…

Computations for prime knots up to 11 crossings.

The $Q$-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the $Q$-prime curvature under scaling is given in terms of a differential operator, called the $P$-prime operator, acting on the space of CR pluriharmonic functions. …

New tool lassos connects virtual and surface link diagrams, changing primeness rules.

The paper proves a generalized Kauffman-Harary conjecture for prime determinant links.

We show that the Frölicher spectral sequence of a complex parallelizable solvmanifold is degenerate at $E_{2}$-term. For a semi-direct product $G=\C^{n}\ltimes_φN$ of Lie-groups with lattice $Γ=Γ^{\prime}\ltimes Γ^{\prime\prime}$ such that $N$ is a nilpotent Lie-group with a left-invariant complex structure and $φ$ is …

Study L^2-Betti numbers in prime characteristic for a conjecture about 2-complex towers.

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an…

Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in $S^3$. We prove a similar result for links in closed thickened surfaces $S \times I$. We define a link to be fully alternating if it has an alternating projection from $S\times I$ to $S$ where the interior of every complemen…

The paper finds at least four prime closed characteristics on star-shaped hypersurfaces in 8D space.

Study counts geodesics on hyperbolic 3-manifolds, proving prime theorems.

Decomposes string links in a surface into prime components.

We construct prime amphicheiral knots that have free period 2. This settles an open question raised by the second named author, who proved that amphicheiral hyperbolic knots cannot admit free periods and that prime amphicheiral knots cannot admit free periods of order >2.

Prime homology detects split links in prime characteristic.

Proves rigidity of SU(2) and SO(3) quantum representations at prime levels.

pPCA speeds up PCA by priming initial estimates for faster, more accurate results.

Researchers create higher-dimensional $\mathcal{I}^\prime$-curvatures and find counterexamples to the Hirachi conjecture.

Identifies doubly slice genera for 2909 prime knots with up to 12 crossings.

This paper finds all prime alternating knots with minimal warping degree two.

This is the second of a part series devoted to enumerating prime alternating knots and links. In Part I, we introduced four operators on knots and showed that if these operators are applied to the set of all prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossings is obtained. In …

Classifies prime algebraic tangles up to 14 crossings.

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial st…

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial st…

We give an integral formula for the total $Q^\prime$-curvature of a three-dimensional CR manifold with positive CR Yamabe constant and nonnegative Paneitz operator. Our derivation includes a relationship between the Green's functions of the CR Laplacian and the $P^\prime$-operator.

Knots 4_1 and 5_1 can be paired to show unknotting number is not additive.

A partial order on the set of prime knots can be defined by the existence of an epimorphism between knot groups. We prove that all the prime knots with up to $6$ crossings are minimal. We also show that each fibered knot with the irreducible Alexander polynomial is minimal.

Minimal grid diagrams found for 13-crossing prime knots with 13 arc index.

We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-Kähler metric. In the appendix, by Rod Gove…

As a supplement to the authors' article "Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots", to appear in the Journal of Knot Theory and its Ramifications, we present minimal arc presentations of the prime knots up to arc index 11.