纽结的扭曲Kuperberg不变量与通过扭曲Drinfeld双得到的Reidemeister挠率

Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $\text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into $\text{Aut}(H)$ and possibly with a $\text{Spin}^c$ structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $\mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we sh…

平衡缝合三维流形的Kuperberg不变量

Kuperberg invariants for balanced sutured 3-manifolds

本文基于对合Hopf超代数构造带Spin^c结构的平衡缝合三维流形的量子不变量，并证明在Borel子代数情形下该不变量等价于规范化的Reidemeister挠率。

从交换霍普夫代数中获得3流形与嵌入带框架链接的不变量

Invariants from involutory Hopf algebras of 3-manifolds with embedded framed links

本文基于交换霍普夫代数和Drinfeld双代数定义了3流形与嵌入链接的不变量，推广了Kuperberg和HKR不变量，并通过Heegaard-Link图示方法构建了新的证明。

Drinfeld-Manin solutions of the Yang-Baxter equation coming from cube complexes

The most common geometric interpretation of the Yang-Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. We will show that there are higher-dimensional cube complexes solving the $D$-state Yang-Baxter equation for arbitrarily large $D$. More precisely, we introduce explicit constructions of cube complexes covered by products of $n$ trees and show that these cube complexes lead to new solutions of the Yang-Baxter equations.

On Two Invariants of Three Manifolds from Hopf Algebras

We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-Westbury $(\text{TVBW})$ invariant and the Witten-Reshetikhin-Turaev $(\text{WRT})$ invariant, respectively. By a classical result relating $\text{TVBW}$ and $\text{WRT}$, it follows that the Kuperberg invariant for a semisimple Hopf algebra is equal to the Hennings-Kauffman-Radfo…

The universal quantum invariant and colored ideal triangulations

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal $R$-matrix. On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has…