关于单位根上的Burau表示

量子群$\mathcal{U}_ξ\mathfrak{sl}(2|1)$在单位根上的拓扑不变量

Topological invariants from quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ at roots of unity

本文从李超代数$\mathfrak{sl}(2|1)$的量子群出发，构造了链不变量和3-流形不变量。这种构造基于量子群$\mathcal{U}_ξ\mathfrak{sl}(2|1)$的幂零不可约有限维表示，其中$ξ$是奇数阶的单位根。这些构造使用了修改迹和相对$\mathit{G}$ -模范畴的概念。

Steinberg skein identities at roots of unity

We obtain a family of skein identities in the Kauffman bracket skein module which relate Frobenius elements to Jones-Wenzl projectors at roots of unity. We view these skein identities as certain incarnations of Steinberg tensor product formulae from the theory of tilting modules of the quantum group $U_q(sl_2)$. We show that the simplest skein identities yield a short new proof of the existence of the Chebyshev-Frobenius homomorphism of Bonahon-Wong.

单位根上的可估Jones-Wenzl幂等元与U_qsl（2）中心上的模表示

Evaluable Jones-Wenzl idempotents at root of unity and modular representation on the center of U_qsl(2)

设p为整数。本文在Temperley - Lieb代数的4p阶单位根上定义了一族幂等元（和幂零元），推广了通常的Jones-Wenzl幂等元.这些新的幂等元对应于受限量子群Uqsl（2）的有限维单射影不可分解表示，其中q是2 p阶单位根.在[BHMV 95]拓扑量子场论（TQFT）的方式，他们提供了一个规范的基础上，在彩色绞链模块定义映射类群表示。在环面情况下，这个基允许我们获得负扭转和屈曲作用之间的部分匹配，以及在Uqsl（2）中心的SL 2（Z）的[LM 94]诱导表示，它非平凡地扩展了SL 2（Z）的[RT 91]表示。

素数单位根处的有色 Khovanov 同调理论

Colored Khovanov homology theories at prime roots of unity

本文为有色 Khovanov 同调引入次数 2 导子，在特征 p>2 域上得到 p-幂零微分，实现 2p 次单位根处 gl(2) 同调的范畴化。

四阶单位根上的Skein模

Skein modules at the 4th roots of unity

对于所有的3-流形M，当A=+1，-1时，通过把它们与SL（2，C）-特征标簇联系起来，计算了Kauffman括号skein模S（M，A）.我们将这种描述扩展到当A是1的4次根且M是曲面x [0，1]或有理同调球面（或其子流形）的情况。

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.

单位根上$\textbf{U}_q$ -倾斜模的图范畴

Diagram categories for $\textbf{U}_q$-tilting modules at roots of unity

我们给出了一个图形表示的范畴的$\textbf{U}_q(\mathfrak{sl}_2)$ -倾斜模块$\mathfrak{T}$的$q$是一个单位根，并介绍了一个分级上$\mathfrak{T}$。这种分级是“统一现象的根源”，并可能导致新的见解链接和$3$ -流形不变量推导出$\mathfrak{T}$。我们也给了一个图解类别的（分级）投射endofunctors在$\mathfrak{T}$，表明我们的结果可以推广和收集一些“众所周知的”事实，给一个合理的自足的论述。

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。

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.

Asymptotics of Nahm sums at roots of unity

We give a formula for the radial asymptotics to all orders of the special $q$-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in~\cite{CGZ} to prove one direction of Nahm's conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in $K$-theory. The power series occurring in our asymptotic formula are identical to the conjectured asymptotics of the Kashaev invariant of a knot once we convert Neumann-Zagier data into Nahm data, suggesting a deep connection between asymptotics of quantum knot invariants and asymptotics of Nahm…