相关论文: Quantum Knot Invariant for Torus Link and Modular …

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in $S^3$. Via…

In 2003, Hikami and Kirillov uncovered an intriguing connection between torus knots $\mathcal{K}_{(P,Q)}$ and Virasoro minimal models $\mathcal{M}(P,Q)$ by relating the Kashaev invariants of the knots to the characters of the corresponding…

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra $\mathfrak{sl}(2|1)$. This construction based on nilpotent irreducible finite dimensional representations of…

Echeverria recently introduced an invariant for a smoothly embedded torus in a homology $S^1\times S^3$, using gauge theory for singular connections. We define a new topological invariant of such an embedded torus, analogous to the…

From analysis of a big variety of different knots we conclude that at q which is an root of unity, q^{2m}=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: H_{r+m} = H_r H_m for any A, which is a…

We construct a new family of knot concordance invariants $\theta^{(q)}(K)$, where $q$ is a prime number. Our invariants are obtained from the equivariant Seiberg-Witten-Floer cohomology, constructed by the author and Hekmati, applied to the…

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…

The Kashaev-Murakami-Murakami Volume Conjecture connects the hyperbolic volume of a knot complement to the asymptotics of certain evaluations of the colored Jones polynomials of the knot. We introduce a closely related conjecture for…

We report on recent results of the authors concerning calculations of quantum invariants of Seifert 3-manifolds. These results include a derivation of the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated with an…

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize…

We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman…

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial evaluated at $\exp(\xi/N)$ for a real number $\xi$ greater than a certain constant. We prove that, from the asymptotic behavior, we can extract the…

The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic…

Using the Racah coefficients in our earlier paper arXiv:1107.3918, we explicitly write the Chern-Simons field theory invariants for many non-torus knot and links. Further, we have tabulated the reformulated invariants which agrees with the…

We construct {\it quantum hyperbolic invariants} (QHI) for triples $(W,L,\rho)$, where $W$ is a compact closed oriented 3-manifold, $\rho$ is a flat principal bundle over $W$ with structural group $PSL(2,\mc)$, and $L$ is a non-empty link…

The category of finite dimensional module over the quantum superalgebra U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even when q is not a…

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up…

Based on different views on the Jones polynomial we review representation theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them via the theory of Soergel bimodules. The…

Recently, Mullins calculated the Casson-Walker invariant of the 2-fold cyclic branched cover of an oriented link in S^3 in terms of its Jones polynomial and its signature, under the assumption that the 2-fold branched cover is a rational…

We extend the definition of the U(1)-reducible connection contribution to the case of the Witten-Reshetikhin-Turaev invariant of a link in a rational homology sphere. We prove that, similarly ot the case of a link in S^3, this contribution…