arXiv research

Modified Hennings invariant defined using quantum groups and integrals.

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with Jones- Witten-Reshetikhin-Turaev invariant arising from Chern-Simons filed theory for the same Lie algebra and the same root of unity on all integer homol- ogy three-spheres, at roo…

We prove the unrolled superalgebra $\mathcal{U}_ξ^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $ξ$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of…

We construct non-semisimple $2+1$-TQFTs yielding mapping class group representations in Lyubashenko's spaces. In order to do this, we first generalize Beliakova, Blanchet and Geer's logarithmic Hennings invariants based on quantum $\mathfrak{sl}_2$ to the setting of finite-dimensional non-degenerate unimodular ribbon H…

Paper connects two invariants of 3D manifolds using Hopf algebras.

Invariants for 4-manifolds from Hopf group-algebras.

We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold $M$ and a colored link $L$ inside $M$. The link $L$ is split into two parts colored…

We provide a general construction of integral TQFTs over a general commutative ring, $\mathbf{k}$, starting from a finite Hopf algebra over $\mathbf{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds. We show the construction app…

We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-W…

M. Hennings and G. Kuperberg defined quantum invariants Z_{Henn} and Z_{Kup} of closed oriented 3-manifolds based on certain Hopf algebras, respectively. We prove that |Z_{Kup}|=|Z_{Henn}|^2 for lens spaces when both invariants are based on factorizable finite dimensional ribbon Hopf algebras.

We show that unrolled quantum groups at odd roots of unity give rise to relative modular categories. These are the main building blocks for the construction of 1+1+1-TQFTs extending CGP invariants, which are non-semisimple quantum invariants of closed 3-manifolds decorated with ribbon graphs and cohomology classes. Whe…

We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…

We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum…

Hennings and Kuperberg defined quantum invariants $Z_{Henn}$ and $Z_{Kup}$ for closed oriented $3$-manifolds based on certain Hopf algebras, respectively. When the Hopf algebras are semisimple, it is shown that $Z_{Kup}=|Z_{Henn}|^2$. In this paper, we present a new proof of this equality.

Quantum invariants for surfaces in 4D 2-handlebodies.

This paper updates knot invariants using Hopf algebras and categorifies their structure.

We extend the construction of the Hennings TQFT for ribbon Hopf algebras to the case of ribbon quasi-Hopf algebras as defined by Drinfeld. Calculations proceed in a similar fashion to the ordinary Hopf algebra case, but also require the handling of the non-trivial coassociator in the triple tensor product of the algebr…

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…

New (3+1) TQFTs created from non-semisimple categories.

Invariants for 3-manifolds with embedded links using Hopf algebras.

Constructs TQFTs for cobordisms with cohomology class decorations.

T-BFA targets and misleads specific DNN inputs to a chosen output.

Abstract invariant cannot be expressed using various slice-torus invariants.

The $θ$ invariant encompasses the Rozansky-Overbay invariant.

Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.

Study of Bauer-Furuta invariants under Lie group actions and Galois coverings.

The Kuperberg invariant is shown to be gauge invariant for certain framed 3-manifolds.

New polynomial invariant distinguishes singular links.

We show that the perturbative ${\frak g}$ invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra ${\frak g}$, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …

Paper introduces new invariant for pairs of immersions.

Grid homology confirms the Upsilon invariant in knot theory.

Constructs universal link invariants from intersections in configuration spaces.

New invariant $CWR$ for alternating links is stronger than existing invariants.

Defines knot concordance invariant using instanton homology and Donaldson invariants.

Combines combinatorial method to extend Milnor invariants to welded links.

New family of knots with epsilon invariant nonzero despite Upsilon and phi being zero.

Paper introduces a new invariant for virtual knotoids and proves it's a Vassiliev invariant of order one.

We construct two knot invariants. The first knot invariant is a sum constructed using linking numbers. The second is an invariant of flat knots and is a formal sum of flat knots obtained by smoothing pairs of crossings. This invariant can be used in conjunction with other flat invariants, forming a family of invariants…

As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich. We show that the Orr invariant of degree $k$ is equivalent to the tree reduction of the Kontsevich invariant of degree $< 2k$. Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and …

Formula connects surface and curve invariants via slice transitions.

In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called $β$-invar…

New concordance invariants phi and phi_j are defined and studied.

We recall the definition of the quadratic helicity invariant and of the higher asymptotic ergodic $M$-invariant. We present a simpler new proof (in part) that the $M$-invariant is ergodic. The $M$-invariant is a higher invariant, this means that for the magnetic field with closed magnetic lines the invariant is not a f…

Defines new link-homotopy invariants using Milnor's higher order link invariants.

New invariants for singular knots and links defined using shadow structures.

We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, $ρ$-invariants and $String$-bordism invariants are derived as special cases. The main results are a secondary index theo…

Paper calculates L-invariant and L*-invariant for complex surface sums.

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