Related papers: A unified Casson-Lin invariant for the real forms …

We show some computations on representations of the fundamental group in SL(2;C) and Reidemeister torsion for a homology 3-sphere obtained by Dehn surgery along the figure-eight knot. This is the second version. We recorrected several…

We introduce a diffeomorphism invariant of $4$-manifolds, the $\mathrm{Pin}^-(2)$-monopole invariant, defined by using the $\mathrm{Pin}^-(2)$-monopole equations. We compute the invariants of several $4$-manifolds, and prove gluing…

For a compact 2-orbifold with negative Euler characteristic $\mathcal O^2$, the variety of characters of $\pi_1(\mathcal O^2)$ in $\mathrm{SL}_{n}(\mathbb R)$ is a non-singular manifold at $\mathbb C$-irreducible representations. In this…

The Rankin--Cohen brackets provide a basic example of ``non-elementary" differential symmetry breaking operators. They can be interpreted as bi-differential operators remarkable for reflecting the structure of fusion rules for holomorphic…

We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labelled by irreducible representations of U_q(sl(2)). We show that the corresponding colored invariants of tangles can be…

Let $\mathcal{Z}$ be a spin $4$-manifold carrying a parallel spinor and $M\hookrightarrow \mathcal{Z}$ a hypersurface. The second fundamental form of the embedding induces a flat metric connection on $TM$. Such flat connections satisfy a…

Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an…

The asymptotic expansion of quantum knot invariants in complex Chern-Simons theory gives rise to factorially divergent formal power series. We conjecture that these series are resurgent functions whose Stokes automorphism is given by a pair…

Let $\mathcal {M}$ be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let $cl(\Sigma^{(1)}_{iness})$ be the closure of the union of all singular knots in $\mathcal…

We introduce an invariant of a hyperbolic knot which is a map $\alpha\mapsto \boldsymbol{\Phi}_\alpha(h)$ from $\mathbb{Q}/\mathbb{Z}$ to matrices with entries in $\overline{\mathbb{Q}}[[h]]$ and with rows and columns indexed by the…

We produce combinatorial formulas for invariants of smooth embeddings of $(2\ell-1)$-spheres into $\mathbb{R}^{3\ell}$ for $\ell\geq 2$. Furthermore, we obtain such a formula for the Haefliger invariant, which classifies smooth knots…

It is known that the fundamental group homomorphism $\pi_1(T^2) \to \pi_1(S^3\setminus K)$ induced by the inclusion of the boundary torus into the complement of a knot $K$ in $S^3$ is a complete knot invariant. Many classical invariants of…

Using Penrose binor calculus for $SU(2)$ ($SL(2,C)$) tensor expressions, a graphical method for the connection representation of Euclidean Quantum Gravity (real connection) is constructed. It is explicitly shown that: {\it (i)} the recently…

An invariant for twisted K theory classes on a 3-manifold is introduced. The invariant is then applied to the twisted equivariant classes arising from the supersymmetric Wess-Zumino-Witten model based on the group SU(2). It is shown that…

We give a spinorial representation of a submanifold of any dimension and co-dimension in a symmetric space $G/H,$ where $G$ is a complex semi-simple Lie group and $H$ is a compact real form of $G.$ This in particular includes…

In view of the self-linking invariant, the number $|K|$ of framed knots in $S^3$ with given underlying knot $K$ is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that $|K|$ is…

Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting…

We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU(3)xSU(3) transformation group are used to establish the count of algebraically independent polynomial…

In this paper we define and investigate Z/2-homology cobordism invariants of Z/2-homology 3-spheres which turn out to be related to classical invariants of knots. As an application we show that many lens spaces have infinite order in the…

The traceless $SU(2)$ character variety $R(S^2,\{a_i,b_i\}_{i=1}^n)$ of a $2n$-punctured 2-sphere is the symplectic reduction of a Hamiltonian $n$-torus action on the $SU(2)$ character variety of a closed surface of genus $n$. It is…