Related papers: Quantum Knot Invariant for Torus Link and Modular …
The Links-Gould invariant of links $LG^{2,1}$ is a two-variable generalization of the Alexander-Conway polynomial. Using representation theory of $U_{q}\mathfrak{gl}(2 \vert 1)$, we prove that the degree of the Links-Gould polynomial…
The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at…
We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a…
We show that the differences between various concordance invariants of knots, including Rasmussen's $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are…
In this article, we obtain an asymptotic expansion formula for the relative Reshetikhin-Turaev invariant in the case that the ambient 3-manifold is gained by doing rational surgery along one component of Whitehead link. In addition, we…
Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant $t^{I\left( \mathcal{L} \right) }$ is constructed for a link $\mathcal{L}$, where $I$ is the abelian Chern-Simons…
Using the fiber bundle concept developed in geometry and topology, the fractionally quantized Hall conductivity is discussed in the relevant many--particle configuration space. Electron-magnetic field and electron-electron interactions…
We prove that a formal power series associated to an ideally triangulated cusped hyperbolic 3-manifold (together with some further choices) is a topological invariant. This formal power series is conjectured to agree to all orders in…
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 show that the number of homomorphisms from a knot group to a finite group $G$ cannot be a Vassiliev invariant, unless it is constant on the set of $(2,2p+1)$ torus knots. In several cases, such as when $G$ is a dihedral or symmetric…
Details of quantum knot invariant calculations using a specific SU(3)_q-module are given which distinguish the Conway and Kinoshita-Teresaka pair of mutant knots. Features of Kuperberg's skein-theoretic techniques for SU(3)_q invariants in…
Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p,…
We prove an explicit formula for the tail of the colored Jones polynomial for a class of arborescent links in terms of a product of theta functions and/or false theta functions. We also provide numerical evidence towards a classification of…
In this paper, we extend the definition of a knotoid that was introduced by Turaev, to multi-linkoids that consist of a number of knot and knotoid components. We study invariants of multi-linkoids that lie in a closed orientable surface,…
Ropelength, L, is a parameter characterizing the minimum contour length of a knot or link. There exist upper and lower bounds on ropelength with respect to crossing number, C, including a universal lower bound constraining $L\geq\alpha_0…
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 Mazorchuk-Stroppel…
The modular data of a modular category $\mathcal{C}$, consisting of the $S$-matrix and the $T$-matrix, is known to be an incomplete invariant of $\mathcal{C}$. More generally, the invariants of framed links and knots defined by a modular…
We introduce the concept of tied links in the solid torus, which generalize naturally the concept of tied links in $S^3$ previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein…
We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M,L),…
A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $\omega q$…