arXiv research

The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we exhibit explicit relationships between the universal sl_2 invariant and Milnor invaria…

We construct an invariant for non-spin 4-manifolds by using 2-torsion cohomology classes of moduli spaces of instantons on SO(3)-bundles. The invariant is an SO(3)-version of Fintushel-Stern's 2-torsion instanton invariant. We show that this SO(3)-torsion invariant is non-trivial for $2CP^2 # -CP^2$, while it is known …

Survey of invariants for knotted 2-spheres in 4-space.

Survey on L^2-invariants for 3-manifolds.

Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.

Researchers compute mod 2 Seiberg-Witten invariants for spin structures and families.

The paper introduces a new invariant for 2-knots in S^4.

Established a stable cohomotopy refinement for Pin(2) monopole invariants.

Defines a new $SL_2(\mathbb{R})$-Casson invariant using Reidemeister torsions.

Researchers found a family of Sp(2)-invariant solitons for Laplacian flow.

New invariants detect exotic smooth structures in 4-manifolds.

This paper classifies 2-plat 2-knots using a new invariant.

We explore a knot invariant derived from colorings of corresponding $1$-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle $2$-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles w…

New rational curvature measures for 2-complexes.

For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and at any even …

Study on Casson invariant and its variants in mapping class groups.

The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1- and 2- handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an elem…

Study $G_2$-instantons on $\mathbb{R}^4 imes S^3$ with invariant properties.

Researchers derive $q$-series for $SO(3)$ and $OSp(1|2)$ groups.

We use intersection theory techniques to define an invariant of closed 3-manifolds counting the characters of irreducible representations of the fundamental group in PSL(2,C). We note several properties of the invariant and compute the invariant for certain Seifert fibered spaces and for some Dehn surgeries on twist kn…

Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.

Two new polynomial invariants for long virtual knots.

C. Giller proposed an invariant of ribbon 2-knots in S^4 based on a type of skein relation for a projection to R^3. In certain cases, this invariant is equal to the Alexander polynomial for the 2-knot. Giller's invariant is, however, a symmetric polynomial -- which the Alexander polynomial of a 2-knot need not be. Afte…

Constructs instantons on a specific G_2-manifold limit.

New formulas for spatial 2-bouquet graphs discovered.

Study coclosed G2-structures on SU(2)²-invariant manifolds.

Two specific Einstein metrics found on a product of SL(2,R) groups.

Quantum invariants from $U_h\mathfrak{sl}(2|1)$ are q-holonomic.

The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.

Complete invariant defined for doodles on a sphere.

This research studies affine invariance in continuous-domain convolutional neural networks.

This paper deals with the study of a new family of knot invariants: the $L^2$-Alexander invariant. A main result is to give a method of computation of the $L^2$-Alexander invariant of a knot complement using any presentation of default 1 of the knot group.

Computes a $ν$-invariant for Joyce's $G_2$-manifolds.

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group then th…

Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.

In the spirit of [10,2], we study the Calabi-Yau equation on $T^2$-bundles over $\mathbb{T}^2$ endowed with an invariant non-Lagrangian almost-Kähler structure showing that for $T^2$-invariant initial data it reduces to a Monge-Ampère equation having a unique solution. In this way we prove that for every total space $M…

New invariants for 4-manifolds obstruct certain surface pairs.

Proves simple type conjecture for mod 2 Seiberg-Witten invariants.

The paper studies $G_2$-Poisson equations on 7-spheres and classifies invariant solutions.

The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)…

Cochran, Orr and Teichner introduced $L^2$--eta--invariants to detect highly non--trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian $L^2$--eta--invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction t…

We show that the L^2-torsion and the von Neumann rho-invariant give rise to commensurability invariants of knots.

Study topological Iwasawa invariants for 3-sphere links, proving density results.

We consider the non-trivial Ricci soliton on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ constructed by Koiso and Cao. It is a Kähler metric invariant by the $U(2)$ action on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$. We study its Yamabe equation and prove it has exactly one $U(2)-$invariant solution up to homothecies.

The paper studies bundles over surfaces and constructs invariants.

The study finds invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.

We define extensions of the $L^2$-analytic invariants of closed manifolds, called delocalized $L^2$-invariants. These delocalized invariants are constructed in terms of a nontrivial conjugacy class of the fundamental group. We show that in many cases, they are topological in nature. We show that the marked length spect…

Study $d$-invariants of L-space double branched covers of arborescent links.