Valeriano Aiello, Roberto Conti, Vaughan F. R. Jones
We show how to construct unitary representations of the oriented Thompson group $\vec{F}$ from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of $\vec{F}$.
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Positive oriented Thompson links
本文研究 Thompson 群定向子群中正元素所对应链环的性质，证明这些链环均为正链环。
In [Jo14] and [Jo18] Vaughan Jones introduced a construction which yields oriented knots and links from elements of the oriented Thompson group $\vec{F}$. In this paper we prove, by analogy with Alexander's classical theorem establishing that every knot or link can be represented as a closed braid, that given an oriented knot/link $\vec{L}$, there exists an element $g$ in $\vec{F}$ whose closure $\vec{\mathcal{L}}(g)$ is $\vec L$.
On unifying Jones representations of braid groups and Thompson's group
本文引入有向二元辫群，统一了辫群与定向Thompson群的Jones表示，并构造了扩展的酉表示及嵌入，使辫闭包恢复定向链环构造。
Half grid diagrams and Thompson links
本文定义半网格图并证明每个链环可由半网格表示，将其与Jones构造等价，进而用Thompson指标给出纽结最大Thurston-Bennequin数的上下界，并建立半网格图与对称群元素的对应关系。
A few years ago the so-called oriented subgroup $\vec F$ of the Thompson group $F$ was introduced by V. Jones while investigating the connections between subfactors and conformal field theories. In the coding of links and knots by elements of $F$ it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, $\vec F$ provided the first example of a maximal subgroup of infinite index in $F$ different from the parabolic subgroups that fix a point in $(0,1)$. In this paper we investigate possible analogues of $\vec F$ in higher Thompson groups $F_k, k\geq 2$, with $F=F_2$, in…
In a "naive" attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. The Thompson group elements are the "local scale transformations" of the theory. In a simple case the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new "oriented" subgroups $\overrightarrow F <F$ and $\overrightarrow T< T$ which allow us to produce all \emph{oriented} knots and links.
The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $\vec{F}$ and $\vec{T}$ of the Thompson groups $F$ and $T$, respectively, together with a procedure that associates an oriented link diagram to any element of these subgroups. Moreover, several specializations of some well-known polynomial link invariants can be seen as functions of positive type on the Thompson groups or the Jones-Thompson subgroups. One important example is provided by suitable evaluations of the Jones polynomial, …
Thompson's group $F$, tangles, and link homology
本文扩展Jones的构造，将$(n, n)$-缠结与Thompson群$F$元素关联并证明其渐近忠实，进而证明有向Thompson群$\vec F$在Khovanov链复形范畴上存在松弛群作用。
Virtual Thompson's group
本文受虚拟辫群推广自辫群且可生成虚拟链环、Thompson群$F$可构造有向链环的研究启发，定义了Thompson群$F$的虚拟版本并证明其可构造任意虚拟链环。
Thompson's Group $V$ and Virtual Link Theory
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。
Jones introduced unitary representations of Thompson group $F$ starting from a given subfactor planar algebra, and all unoriented links arise as matrix coefficients of these representations. Moreover, all oriented links arise as matrix coefficients of a subgroup $\vec{F}$ which is the stabilizer of a certain vector. Later Golan and Sapir determined the subgroup $\vec{F}$ and showed many interesting properties. In this paper, we investigate into a large class of groups which arises as subgroups of Thompson group $F$ and reveal the relation between the skein theory of the subfactor planar algebr…
Homological stability for the ribbon Higman--Thompson groups
本文通过引入ribbon Higman-Thompson群的概念，证明了其满足同调稳定性，为大映射类群的稠密子群提供了首个同调稳定性结果。
On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$
本文研究 Thompson 群 $F$ 的 $3$-可着色子群 $\mathcal{F}$，证明相关拟正则表示不可约，并构造三个新的无限指数极大子群。
Recently Vaughan Jones showed that the R. Thompson group $F$ encodes in a natural way all knots, and a certain subgroup $\vec F$ of $F$ encodes all oriented knots. We answer several questions of Jones about $\vec F$. In particular we prove that the subgroup $\vec F$ is generated by $x_0x_1, x_1x_2, x_2x_3$ (where $x_i, i=0,1,2,...$ are the standard generators of $F$) and is isomorphic to $F_3$, the analog of $F$ where all slopes are powers of $3$ and break points are $3$-adic rationals. We also show that $\vec F$ coincides with its commensurator. Hence the linearization of the permutational re…
Higman--Thompson groups $F_n$ all the way down
证明对$n\geq2$，希格曼 - 汤普森群$F_n$有无限指数且同构于自身的极大子群。构造相关子群链，刻画了特定同胚，给出将共轭问题转化为算法的机制，并应用于解决一些群同构问题。
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