AI 中文总结
该研究将琼斯关于汤普森群的纽结表示方案推广到汤普森群 $V$，定义了其到加厚曲面CC虚拟纽结的满射，构造了定向子群 $\vec{V}$ 及其酉表示，实现了所有定向几乎经典虚拟纽结。
AI 中文摘要
汤普森群 $F \subset T \subset V$ 于1965年被引入，此后在逻辑、群论、同伦论、格规范理论等多个领域得到广泛应用。2014年，V. F. R. 琼斯构造了 $F$ 的酉表示，该表示通过从 $F$ 到 $S^3$ 中纽结同痕类的满射实现；第二作者将琼斯的满射推广到 $T$，从而构造了加厚环带中所有棋盘可着色（CC）纽结的同痕类。我们完成了针对 $V$ 的该方案，定义了从 $V$ 到加厚紧定向曲面中CC纽结的虚拟等价类的满射 $\mathcal{L}_{V}$，这产生了一个新的定向子群 $\vec{V} \subset V$，其包含琼斯的定向子群 $\vec{F} \subset F$ 和 $\vec{T} \subset T$。我们证明 $\vec{V}$ 实现了所有定向几乎经典虚拟纽结，随后分别从kei和算子 quandles 着色不变量构造了 $V$ 和 $\vec{V}$ 的酉表示。
英文摘要
Thompson's groups $F \subset T \subset V$ were introduced in 1965 and have since found widespread application in fields as diverse as logic, group theory, homotopy theory, and lattice gauge theory. In 2014, V. F. R. Jones constructed unitary representations of $F$, factoring through a surjection from $F$ to isotopy classes of links in $S^3$. The second author extended Jones' surjection to $T$, thereby constructing all isotopy classes of checkerboard colorable (CC) links in the thickened annulus. We complete this program for $V$, defining a surjection $\mathcal{L}_{V}$ from $V$ to virtual equivalence classes of CC links in thickened compact oriented surfaces. This yields a new oriented subgroup $\vec{V} \subset V$ containing Jones' oriented subgroups $\vec{F} \subset F$ and $\vec{T}\subset T$. We prove $\vec{V}$ realizes all oriented almost classical virtual links. We then construct unitary representations of $V$ and $\vec{V}$ from kei and operator quandle coloring invariants, respectively.
Comments36 pages, 41 figures
