Related papers: Quantum trace map for 3-manifolds and a 'length co…

This paper introduces an algebra structure on the part of the skein module of an arbitrary $3$-manifold $M$ spanned by links that represent $0$ in $H_1(M;\mathbb{Z}_2)$ when the value of the parameter used in the Kauffman bracket skein…

Let $K\subset S^3$ be a knot, $X:= S^3\setminus K$ its complement, and $\mathbb{T}$ the circle group identified with $\mathbb{R}/\mathbb{Z}$. To any oriented long knot diagram of $K$, we associate a quadratic polynomial in variables…

Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$…

We give a rational surgery formula for the Casson-Walker invariant of a 2-component link in $S^{3}$ which is a generalization of Matveev-Polyak's formula. As application, we give more examples of non-hyperbolic L-space $M$ such that knots…

Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds,…

In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the $3$-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic…

Recent progress in string theory has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to…

We study skein modules of 3-manifolds by embedding them into the Hilbert spaces of 4d ${\cal N}=4$ super-Yang-Mills theories. When the 3-manifold has reduced holonomy, we present an algorithm to determine the dimension and the list of…

In the prequel of this paper, Kauffman and Ogasa introduced new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed…

We study the quantum modular properties of $\widehat Z{}^G$-invariants of closed three-manifolds. Higher depth quantum modular forms are expected to play a central role for general three-manifolds and gauge groups $G$. In particular, we…

In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular…

We consider the link and three-manifold invariants from arXiv:1912.02063, which are defined in terms of certain non-semisimple finite ribbon categories $\mathcal{C}$ together with a choice of tensor ideal and modified trace. If the ideal is…

We introduce the notion of modular $q$-holonomic modules whose fundamental matrices define a cocycle with improved analyticity properties and show that the generalised $q$-hypergeometric equation, as well as three key $q$-holonomic modules…

We investigate some particular completely positive maps which admit a stable commutative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a…

We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.

The skein algebra of an oriented $3$-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the $SL_2(\mathbb{C})$-character variety. In this paper we determine the quotient of a polynomial ring…

It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the…

We study a quantum version of the $n$-dimer model from statistical mechanics, based on the formalism from quantum topology developed by Reshetikhin and Turaev (the latter which, in particular, can be used to construct the Jones polynomial…

We review some recent developments in Chern-Simons theory on a hyperbolic 3-manifold $M$ with complex gauge group $G$. We focus on the case $G=SL(N,\mathbb{C})$ and with $M$ a knot complement. The main result presented in this note is the…

An invariant $\mu_{\alpha}(K)$ of fibred knots $K$ in a homology sphere is defined for each $\alpha \in {\bold S}{\bold U}_n$ as follows. Since the knot is fibred, the knot complement is described by an element of the mapping class group,…