A rationality conjecture about Kontsevich integral of knots and its implications to the structure of the colored Jones polynomial

The lines of the Kontsevich integral and Rozansky's rationality conjecture

This work develops some technology for accessing the loop expansion of the Kontsevich integral of a knot. The setting is an application of the LMO invariant to certain surgery presentations of knots by framed links in the solid torus. A consequence of this technology is a certain recent conjecture of Rozansky's. Rozansky conjectured that the Kontsevich integral could be organised into a series of ``lines'' which could be represented by finite $\Qset$-linear combinations of diagrams whose edges were labelled, in an appropriate sense, with rational functions. Furthermore, the conjecture requires…

The colored Jones polynomial and Kontsevich-Zagier series for double twist knots, II

Let $K_{(m,p)}$ denote the family of double twist knots where $2m-1$ and $2p$ are non-zero integers denoting the number of half-twists in each region. Using a result of Takata, we prove a formula for the colored Jones polynomial of $K_{(-m,-p)}$ and $K_{(-m,p)}$. The latter case leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. We also use Bailey pairs and formulas of Walsh to find cyclotomic-like expansions for the colored Jones polynomials of $K_{(m,p)}$ and $K_{(m,-p)}$.

双扭结的着色Jones多项式与Kontsevich-Zagier级数

The colored Jones polynomial and Kontsevich-Zagier series for double twist knots

本文利用Takata的结果，证明了双扭结$K_{(-m,-p)}$和$K_{(-m,p)}$的着色Jones多项式公式，并由此导出新的$q$-超几何级数族，推广了Kontsevich-Zagier级数及其与强单峰序列生成函数在单位根处的对偶性。

Beads: From Lie algebras to Lie groups

The Kontsevich integral of a knot is a powerful invariant which takes values in an algebra of trivalent graphs with legs. Given a Lie algebra, the Kontsevich integral determines an invariant of knots (the so-called colored Jones function) with values in the symmetric algebra of the Lie algebra. Recently A. Kricker and the author constructed a rational form of the Kontsevich integral which takes values in an algebra of trivalent graphs with beads. After replacing beads by an exponential legs, this rational form recovers the Kontsevich integral. The goal of the paper is to explain the relation b…

A universal U(1)-RCC invariant of links and rationality conjecture

We define a graph algebra version of the stationary phase integration over the coadjoint orbits in the Reshetikhin formula for the colored Jones-HOMFLY polynomial. As a result, we obtain a `universal' U(1)-RCC invariant of links in rational homology spheres, which determines the U(1)-RCC invariants based on simple Lie algebras. We formulate a rationality conjecture about the structure of this invariant.

SU(2)/SL(2) knot invariants and KS monodromies

We review the Reshetikhin-Turaev approach to construction of non-compact knot invariants involving R-matrices associated with infinite-dimensional representations, primarily those made from Faddeev's quantum dilogarithm. The corresponding formulas can be obtained from modular transformations of conformal blocks as their Kontsevich-Soibelman monodromies and are presented in the form of transcendental integrals, where the main issue is manipulation with integration contours. We discuss possibilities to extract more explicit and handy expressions which can be compared with the ordinary (compact) …