相关论文: On the volume and the Chern-Simons invariant for t…

In this paper we describe what should perhaps be called a `type-2' Vassiliev invariant of knots S^2 -> S^4. We give a formula for an invariant of 2-knots, taking values in Z_2 that can be computed in terms of the double-point diagram of the…

We present computational data and heuristic arguments which suggest that given a hyperbolic knot the volume correlates with its determinant, the Mahler measure of its Alexander polynomial and the Mahler measure of the twisted Alexander…

We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named…

For a knot K in S^3 we construct according to Casson--or more precisely taking into account Lin and Heusener's further works--a volume form on the SU(2)-representation space of the group of K. We prove that this volume form is a topological…

We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on S^3 (for any representation)…

Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat…

We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schl\"{a}fli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we…

Vassiliev's knot invariants can be computed in different ways but many of them as Kontsevich integral are very difficult. We consider more visual diagram formulas of the type Polyak-Viro and give new diagram formula for the two basic…

In this paper, we extend the symplectic quandle method, previously employed in our study of parabolic representations of knot groups, to investigate the general $SL(2,\mathbb{C})$-representations of 2-bridge ``kmot" groups. We introduce a…

We define a relative version of the Turaev-Viro invariants for an ideally triangulated compact 3-manifold with non-empty boundary and a coloring on the edges, generalizing the Turaev-Viro invariants [35] of the manifold. We also propose the…

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the…

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a $\mathbb{Z}$-cover of a…

We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely…

The Volume conjecture claims that the hyperbolic Volume of a knot is determined by the colored Jones polynomial. The purpose of this article is to show a Volume-ish theorem for alternating knots in terms of the Jones polynomial, rather than…

We consider Conway polynomials of two-bridge links as Euler continuant polynomials. As a consequence, we obtain new and elementary proofs of classical Murasugi's 1958 alternating theorem and Hartley's 1979 trapezoidal theorem. We give a…

We propose the Volume Conjecture for the relative Reshetikhin-Turaev invariants of a closed oriented $3$-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern-Simons invariant of the…

This short survey, which was written to accompany a minicourse at the BIRS conference "Topology in dimension 4.5", concerns invariants of knotted $2$-spheres in $S^4$, also known as $2$-knots. It covers invariants extracted from the…

We review the string/gauge theory duality relating Chern-Simons theory and topological strings on noncompact Calabi-Yau manifolds, as well as its mathematical implications for knot invariants and enumerative geometry.

We use the Chern-Simons quantum field theory in order to prove a recently conjectured limitation on the 1/K expansion of the Jones polynomial of a knot and its relation to the Alexander polynomial. This limitation allows us to derive a…

We compute the A-polynomial 2-tuple of twisted Whitehead links. As applications, we determine canonical components of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds.