相关论文: On the volume and the Chern-Simons invariant for t…
In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope $r$ such that $r$-filling of the knot complement has left-orderable…
We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-submodularity of…
We compute the reduced version of Khovanov and Rozansky's sl(N) homology for two-bridge knots and links. The answer is expressed in terms of the HOMFLY polynomial and signature.
For a twist knot $\mathcal{K}_{p'}$, let $M$ be the closed $3$-manifold obtained by doing $(p, q)$ Dehn-filling along $\mathcal{K}_{p'}$. In this article, we prove that Chen-Yang's volume conjecture holds for sufficiently large $|p| + |q|$…
We study the asymptotic behavior of the twisted Alexander polynomial for the sequence of SL(n ,C)-representations induced from an irreducible metabelian SL(2, C)-representation of a knot group. We give the limits of the leading coefficients…
A string link S can be closed in a canonical way to produce an ordinary closed link L. We also consider a twisted closing which produces a knot K. We give a formula for the Conway polynomial of L as a product of the Conway polynomial of K…
We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then…
We elaborate the Chern-Simons field theoretic method to obtain colored HOMFLY invariants of knots and links. Using multiplicity-free quantum 6j-symbols for U_q(sl_N), we present explicit evaluations of the HOMFLY invariants colored by…
We give upper and lower bounds on the leading coefficients of the $L^2$-Alexander torsions of a $3$-manifold $M$ in terms of hyperbolic volumes and of relative $L^2$-torsions of sutured manifolds obtained by cutting $M$ along certain…
A new, formal, non-combinatorial approach to invariants of three-dimensional manifolds of Reshetikhin, Turaev and Witten in the framework of non-perturbative topological quantum Chern-Simons theory, corresponding to an arbitrary compact…
We give parallel constructions of an invariant R(W,f), based on the classical Rogers dilogarithm, and of quantum hyperbolic invariants (QHI), based on the Faddeev-Kashaev quantum dilogarithms, for flat PSL(2,C)-bundles f over closed…
We show that (as conjectured by Lin and Wang) when a Vassiliev invariant of type $m$ is evaluated on a knot projection having $n$ crossings, the result is bounded by a constant times $n^m$. Thus the well known analogy between Vassiliev…
Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.
Previous work of the author and N. Reshetikhin defines an invariant $\operatorname{Z}_{N}^{\psi}(K, \rho, \mu)$ of a knot $K$, a representation $\rho : \pi_{1}(S^{3} \setminus K) \to \operatorname{SL}_2(\mathbb{C})$, and a logarithm $\mu$…
The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister…
Sinha and Vafa \cite {sinha} had conjectured that the $SO$ Chern-Simons gauge theory on $S^3$ must be dual to the closed $A$-model topological string on the orientifold of a resolved conifold. Though the Chern-Simons free energy could be…
In this paper we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle…
A brief review of the development of Chern-Simons gauge theory since its relation to knot theory was discovered in 1988 is presented. The presentation is done guided by a dictionary which relates knot theory concepts to quantum field theory…
In this article we prove a formula for the volume of 4-dimensional polytopes, in terms of their face bivectors, and the crossings within their boundary graph. This proves that the volume is an invariant of bivector-coloured graphs in $S^3$.
In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander…