论文标题
Dunfield-Gukov-Rasmussen猜想的证明
A proof of Dunfield-Gukov-Rasmussen Conjecture
论文作者
论文摘要
2005年，邓菲尔德（Dunfield），古科夫（Dunfield）和拉斯穆森（Rasmussen）猜想了频谱序列的存在，从降低的三分等级的khovanov-rozansky同源性与Ozsváth和Szabó定义的结式浮点同源性结合在一起。本文的主要结果是证明了这种猜想。为此，我们从$ \ mathfrak {gl} _0 $同源性构建了由最后两位作者构建的打结浮子同源性的频谱序列。使用$ \ mathfrak {gl} _0 $同源性配备了降低的三个分级同源性的光谱序列的事实，我们获得了我们的主要结果。第一个光谱序列是Bockstein类型的，来自系数的微妙操纵。主要工具是泡沫和奇异的soergel双模型的量子痕迹，以及$ \ Mathbb z $ - valued Cube的分辨率模型，用于最初由Ozsváth和Szabó建造的，该模型在两个元素上最初由Ozsváth和Szabó建造。作为一个应用程序，我们推断出$ \ mathfrak {gl} _0 $同源性以及降低的三个分级的Khovanov-Rozansky，一个检测到Unnonkot，两个Trefoils，Figure八节和五薄油。
In 2005 Dunfield, Gukov and Rasmussen conjectured an existence of the spectral sequence from the reduced triply graded Khovanov-Rozansky homology of a knot to its knot Floer homology defined by Ozsváth and Szabó. The main result of this paper is a proof of this conjecture. For this purpose, we construct a bigraded spectral sequence from the $\mathfrak{gl}_0$ homology constructed by the last two authors to the knot Floer homology. Using the fact that the $\mathfrak{gl}_0$ homology comes equipped with a spectral sequence from the reduced triply graded homology, we obtain our main result. The first spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules and a $\mathbb Z$-valued cube of resolutions model for knot Floer homology originally constructed by Ozsváth and Szabó over the field of two elements. As an application, we deduce that the $\mathfrak{gl}_0$ homology as well as the reduced triply graded Khovanov-Rozansky one detect the unknot, the two trefoils, the figure eight knot and the cinquefoil.
