论文标题
杨 - 巴克斯特同源性的绞态理论方法
Skein theoretic approach to Yang-Baxter homology
论文作者
论文摘要
我们介绍了对与琼斯多项式相对应的R-MATRIX的Yang-Baxter（YB）同源性和同源组的绞态理论技术。更具体地说，我们表明，琼斯的YB操作员$ r $ for Jones（同源性的标准化）承认绞合分解$ r = i +βα$，其中$α：v^{\ otimes 2} \ rightarrow k $是“杯子”配对的“杯子”配对和$β：k \ rightarrow v^：与$ r $相关的杯子和帽子的水平张量串联可以分解为杯子。我们应用我们的绞线理论方法来确定第二和第三YB同源性组，从而确认了Przytycki和Wang的猜想。此外，我们计算了$ r $的共同体学组，并在更高维度的计算中提供了一些子模块的an灭。
We introduce skein theoretic techniques to compute the Yang-Baxter (YB) homology and cohomology groups of the R-matrix corresponding to the Jones polynomial. More specifically, we show that the YB operator $R$ for Jones, normalized for homology, admits a skein decomposition $R = I + βα$, where $α: V^{\otimes 2} \rightarrow k$ is a "cup" pairing map and $β: k \rightarrow V^{\otimes 2}$ is a "cap" copairing map, and differentials in the chain complex associated to $R$ can be decomposed into horizontal tensor concatenations of cups and caps. We apply our skein theoretic approach to determine the second and third YB homology groups, confirming a conjecture of Przytycki and Wang. Further, we compute the cohomology groups of $R$, and provide computations in higher dimensions that yield some annihilations of submodules.
