相关论文: Hyperbolic Structure Arising from a Knot Invariant
In the generalized topological quantum field theory constructed by Andersen and Kashaev, invariants of 3-manifolds are defined given the combinatorial structure of a tetrahedral decomposition. Furthermore, a variant of the volume conjecture…
We explicitly construct a dynamically incoherent partially hyperbolic endomorphisms of $\mathbb{T}^2$ in the homotopy class of any linear expanding map with integer eigenvalues. These examples exhibit branching of centre curves along…
The fundamental problem of knot theory is to know whether two knots are equivalent or not. As a tool to prove that two knots are different, mathematicians have developed various invariants. Knots invariants are just functions that can be…
A knot complement admits a pseudo-hyperbolic structure by solving Thurston's gluing equations for an octahedral decomposition. It is known that a solution to these equations can be described in terms of region variables, also called…
We explain how to construct certain potential functions for the hyperbolic structures of a knot complement, which are closely related to the analytic functions on the deformation space of hyperbolic structures.
In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of…
The expectation value of Wilson loop operators in three-dimensional SO(N) Chern-Simons gauge theory gives a known knot invariant: the Kauffman polynomial. Here this result is derived, at the first order, via a simple variational method.…
In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the…
By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories…
Link invariants, for 3-manifolds, are defined in the context of the Rozansky-Witten theory. To each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold X. The invariants are evaluated for b_{1}(M)…
Solutions to both the diffeomorphism and the hamiltonian constraint of quantum gravity have been found in the loop representation, which is based on Ashtekar's new variables. While the diffeomorphism constraint is easily solved by…
We generalize the colored Alexander invariant of knots to an invariant of graphs, and we construct a face model for this invariant by using the corresponding 6j-symbol, which comes from the non-integral representations of the quantum group…
In hep-th/9805025, a result for the symmetric 3-loop massive tetrahedron in 3 dimensions was found, using the lattice algorithm PSLQ. Here we give a more general formula, involving 3 distinct masses. A proof is devised, though it cannot be…
New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising…
We describe a new method of producing equations for the canonical component of representation variety of a knot group into $PSL_2(\mathbb{C})$. Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of…
We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural…
Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of…
Using the skew-Hopf pairing, we obtain $\mathcal{R}$-matrix for the two-parameter quantum algebra $U_{v,t}$. We further construct a strict monoidal functor $\mathcal{T}$ from the tangle category $(\mathrm{OTa},\otimes, \emptyset)$ to the…
We study the Vassiliev knot invariant v_2 of degree 2. We present it via the degrees of maps of various configuration spaces related to a knot to products of spheres. This gives rise to numerous geometrical and combinatorial formulas for…
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by…