与量子$\mathfrak{sl}(2)$有关的图在单位根处的不变量

关于单位根上的Burau表示

On Burau representations at roots of unity

考虑由标准生成元的$k$次幂生成的辫群的子群，证明了任意无穷交（与偶$k$）是平凡的.这是由斯奎尔的一些关于布劳在单位根上的辫群表示的核心的描述所激发的。此外，我们还利用几何方法证明了三股辫群在这些表示下的象是有限群（对于少数单位根）或三角群的有限扩张.

Evaluations of link polynomials and recent constructions in Heegaard Floer theory

Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin's "$\mathfrak{sl}(n)$-like" Heegaard Floer knot invariants $HFK_n$ recover both Alexander polynomial evaluations and $\mathfrak{sl}(n)$ polynomial evaluations at certain roots of unity for links in $S^3$. We show that the equality of these evaluations can be viewed as the decategorified content of the conjectured spectral sequences relating $\mathfrak{sl}(n)$ homology and $HFK_n$.

Overview of knot invariants at roots of unity

We discuss different invariants of knots and links that depend on a primitive root of unity. We clarify the definitions of existing invariants with the Reshetikhin-Turaev method, present the generalization of ADO invariants to $\mathcal{U}_q(sl_N)$ and highlight the connections between different invariants.

所有的单位根都由A-多项式检测

All roots of unity are detected by the A-polynomial

对于任意的正整数n，我们构造了无穷多个单尖双曲3-流形，其中每个流形的A-多项式检测每一个单位的n次根.这就回答了库珀、卡勒、吉列、朗和沙伦的一个问题，即统一的根源是以这种方式产生的。

A categorification of the colored Jones polynomial at a root of unity

There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A differential on triply-graded homology discovered by Cautis is compatible with the $p$-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a $2p$th root of unity.

双曲三维流形的理想点的非平凡单位根的例子

Examples of non-trivial roots of unity at ideal points of hyperbolic 3-manifolds

本文给出了SL（2，C）特征簇有理想点且其单位根不为1或-1的双曲三维流形的例子。这就回答了库珀、卡勒、吉莱特、朗和沙伦的一个问题，即除了1和-1之外，是否存在统一的根。

Infinitely many roots of unity are zeros of some Jones polynomials

Let $N=2n^2-1$ or $N=n^2+n-1$, for any $n\ge 2$. Let $M=\frac{N-1}{2}$. We construct families of prime knots with Jones polynomials $(-1)^M\sum_{k=-M}^{M} (-1)^kt^k$. Such polynomials have Mahler measure equal to $1$. If $N$ is prime, these are cyclotomic polynomials $Φ_{2N}(t)$, up to some shift in the powers of $t$. Otherwise, they are products of such polynomials, including $Φ_{2N}(t)$. In particular, all roots of unity $ζ_{2N}$ occur as roots of Jones polynomials. We also show that some roots of unity cannot be zeros of Jones polynomials.

关于单位根上的Kauffman Kauffman Skein模

On Kauffman Bracket Skein Modules at Root of Unity

利用初等skein方法，我们改进和推广了Bonahon和Wong关于根为1的Kauffman括号skein模的中心元和Chebyshev同态的存在性的结果.