Related papers: Non ambiguous structures on 3-manifolds and quantu…
In this paper we construct explicit examples of both closed and non-compact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the co-rank of a 3-manifold group (also known as the cut number) is bounded…
We give parallel constructions of an invariant R(W,f), based on the classical Rogers dilogarithm, and of quantum hyperbolic invariants (QHI), based on the Faddeev-Kashaev quantum dilogarithms, for flat PSL(2,C)-bundles f over closed…
We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable,…
We define a relative version of the Turaev-Viro invariants for an ideally triangulated compact 3-manifold with non-empty boundary and a coloring on the edges, generalizing the Turaev-Viro invariants [35] of the manifold. We also propose the…
In analogy with classical submanifold theory, we introduce morphisms of real metric calculi together with noncommutative embeddings. We show that basic concepts, such as the second fundamental form and the Weingarten map, translate into the…
We investigate the combinatorial analogues, in the context of normal surfaces, of taut and transversely measured (codimension 1) foliations of 3-manifolds. We establish that the existence of certain combinatorial structures, a priori weaker…
The symmetry structure of a quantum field theory is determined not only by the topological defects that implement the symmetry and their fusion rules, but also by the topological networks they can form, which is referred to as the higher…
We propose a general procedure to construct noncommutative deformations of an algebraic submanifold $M$ of $\mathbb{R}^n$, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of $\mathbb{R}^n$, arXiv:2003.03854] valid for…
We decompose linear $\mathrm{G}_2$-structure in canonical ways adapted to 3-dimensional subspaces, in terms of certain natural 1-forms and definite triple of 2-forms, and apply the decompositions to the study of $\mathrm{G}_2$-structure…
Two different constructions of an invariant of an odd dimensional hyperbolic manifold in the K-group $K_{2n-1}(\bar \Bbb Q)\otimes \Bbb Q$ are given. The volume of the manifold is equal to the value of the Borel regulator on that element.…
Non-invertible symmetries in quantum field theory (QFT) generalize the familiar product rule of groups to a more general fusion rule. In many cases, gauged versions of these symmetries can be regarded as dual descriptions of invertible…
In this paper we develop a theory for constructing an invariant of closed oriented 3-manifolds, given a certain type of Hopf algebra. Examples are given by a quantised enveloping algebra of a semisimple Lie algebra, or by a semisimple…
About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed…
We construct quantum invariants of balanced sutured 3-manifolds with a $Spin^{c}$ structure out of an involutive (possibly non-unimodular) Hopf superalgebra $H$. If $H$ is the Borel subalgebra of $U_{q}(\mathfrak{gl}(1|1))$, we show that…
For a point $p$ in a smooth real hypersurface $M\subset\C^n$, where the Levi form has the nontrivial kernel $K^{10}_p$, we introduce an invariant cubic tensor $\tau^3_p \colon \C T_p \times K^{10}_p \times \overline{K^{10}_p} \to \C\otimes…
Given a fibred hyperbolic 3-manifold with boundary, we coarsely relate the Euclidean geometry of its cusps to the classical fractional Dehn twist coefficient of its monodromy. This result fits into the broader programme of coarsely…
Symmetry plays an important role in the topological band theory. In contrary, study on the topological properties of the asymmetric systems is rather limited, especially in higher-dimensional systems. In this work, we explore a new theory…
We construct a combinatorial invariant of 3-orbifolds with singular set a link that generalizes the Turaev torsion invariant of 3-manifolds. We give several gluing formulas from which we derive two consequences. The first is an…
A twisted-austere $k$-fold $(M, \mu)$ in $\mathbb R^n$ consists of a $k$-dimensional submanifold $M$ of $\mathbb R^n$ together with a closed $1$-form $\mu$ on $M$ such that the `twisted conormal bundle' $N^* M + \mu$ is a special Lagrangian…
Mixed 3-structures are odd-dimensional analogues of paraquaternionic structures. They appear naturally on lightlike hypersurfaces of almost paraquaternionic hermitian manifolds. We study invariant and anti-invariant submanifolds in a…