论文标题
关于彩色琼斯多项式的两个状态模型的几何形状
On the geometry of two state models for the colored Jones polynomial
论文作者
论文摘要
使用定义有色琼斯多项式的R-Matrix的流量特性，我们在链接投影的部分上建立了一组状态之间的自然培训，而在相应的两分之一的二十片上，称为Arc-Graph，称为Arc-Graph，由Garoufalidis和Loebl定义。我们使用它为Garoufalidis和Loebl的打结状态和结节公式提供了新的基本证明。我们将表明，根据我们的二组状态，$ u_q（sl（2，\ mathbb {c}））$ u_q（sl（2，\ mathbb {c}））定义的状态和状态的状态和状态贡献是根据我们的一组状态的培训，对Garoufalidis和Loebl的贡献的贡献。这将表明，这两个状态模型实际上并非基本不同。我们的方法还将将Garoufalidis和Loebl的公式扩展到链接。这需要一些有关部分弧形图的状态几何形状的其他非平凡观察。我们将详细讨论Arc-Graph State-sum的计算，尤其是针对3编封码的封闭。
Using the flow property of the R-matrix defining the colored Jones polynomial, we establish a natural bijection between the set of states on the part arc-graph of a link projection and the set of states on a corresponding bichromatic digraph, called arc-graph, as defined by Garoufalidis and Loebl. We use this to give a new and essentially elementary proof for a knot state-sum formula of Garoufalidis and Loebl. We will show that the state-sum contributions of states on the part arc-graph defined by the universal R-matrix of $U_q(sl(2,\mathbb{C}))$ correspond, under our bijection of sets of states, to the contributions in the formula of Garoufalidis and Loebl. This will show that the two state models are in fact not essentially distinct. Our approach will also extend the formula of Garoufalidis and Loebl to links. This requires some additional non-trivial observations concerning the geometry of states on part arc-graphs. We will discuss in detail the computation of the arc-graph state-sum, in particular for 3-braid closures.
