Quantum link homology via trace functor I
Quantum link homology via trace functor I
复制标题
通过迹函子 I 的量子链接同源性
DOI：
10.1007/s00222-018-0830-0
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发表时间：
2016
影响因子：
3.1
通讯作者：
S. Wehrli
中科院分区：
文献类型：
--
作者：
A. Beliakova;K. Putyra;S. Wehrli
在拓扑学的激励下，我们发展了一个关于内分类的一般理论，它是一对：双分类和内双分类。对于一个梯度线性双范畴和一个固定可逆参数q，我们使用内函子$$\Sigma _q$$ Σq量化该理论，使得对于任意2-态射$$\alpha $$ α, $$\Sigma _q \alpha :=q^{-\deg \alpha }\Sigma \alpha $$ Σqα:=q-degαΣα，否则与$$\Sigma $$ Σ重合。将量子化迹应用于Chen-Khovanov双模的二范畴，得到了一种新的三次分级连杆同调理论，称为量子环连杆同调。如果$$q=1$$ q=1，我们重现了增厚环中链接的Asaeda-Przytycki-Sikora同源性。我们证明了我们的同调带有一个作用，它交织着协调的作用。特别地，n-电缆的量子环同源性承认编织群的作用，编织群通过琼斯串关系与量子群的作用和因子交换。这产生了一个在四维中打结的曲面的非平凡不变量。此外，对环面链路的直接计算表明，量子环同调群的秩依赖于量子参数q。
Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a pair: bicategory and endobifunctor . For a graded linear bicategory and a fixed invertible parameter q, we quantize this theory by using the endofunctor $$\Sigma _q$$Σq such that $$\Sigma _q \alpha :=q^{-\deg \alpha }\Sigma \alpha $$Σqα:=q-degαΣα for any 2-morphism $$\alpha $$α and coincides with $$\Sigma $$Σ otherwise. Applying the quantized trace to the bicategory of Chen–Khovanov bimodules we get a new triply graded link homology theory called quantum annular link homology. If $$q=1$$q=1 we reproduce Asaeda–Przytycki–Sikora homology for links in a thickened annulus. We prove that our homology carries an action of , which intertwines the action of cobordisms. In particular, the quantum annular homology of an n-cable admits an action of the braid group, which commutes with the quantum group action and factors through the Jones skein relation. This produces a nontrivial invariant for surfaces knotted in four dimensions. Moreover, a direct computation for torus links shows that the rank of quantum annular homology groups depend on the quantum parameter q.