论文标题
米尔诺（Milnor）的一致性不变性在表面上
Milnor's concordance invariants for knots on surfaces
论文作者
论文摘要
Milnor的$ \barμ$ $ invariants的链接中的$ 3 $ -sphere $ s^3 $在与边界链接一致的任何链接上都消失了。特别是，它们在$ s^3 $中的任何结上都是微不足道的。在这里，我们考虑在增厚的表面上的结$σ\ times [0,1] $，其中$σ$关闭并定向。我们通过在$ s^3 $中调整陈 - 米尔诺（Chen-Milnor）的链接理论来构建新的一致性不变性，以扩展虚拟结的群体。关键成分是bar-natan $ \ textit {zh} $映射，它允许对组扩展的几何解释。该小组扩展本身最初是由银威廉姆斯定义的。我们的延长$ \barμ$ invariants会阻碍一致性在增厚的表面上的同源结。我们使用它们为具有琐碎的rasmussen不变性，分级属，仿射指数（或Writhe）多项式和广义亚历山大多项式的非斜切虚拟结的新示例。此外，我们完成了所有虚拟结的切片状态分类，最多五个经典的交叉口，并减少到4个（在92800中）的虚拟结数量多达六个具有未知切片状态的经典交叉口。 我们的主要应用程序是Turaev的一致性组$ \ MATHSCR {VC} $在表面上。 Boden和Nagel证明了一致性组的$ \ Mathscr {C} $在$ s^3 $中嵌入$ \ Mathscr {vc} $的中心。与古典结一致性小组相反，我们显示$ \ mathscr {vc} $不是Abelian；回答Turaev提出的问题。
Milnor's $\barμ$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our extended $\barμ$-invariants obstruct concordance to homologically trivial knots in thickened surfaces. We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index (or writhe) polynomial, and generalized Alexander polynomial. Furthermore, we complete the slice status classification of all virtual knots up to five classical crossings and reduce to four (out of 92800) the number of virtual knots up to six classical crossings having unknown slice status. Our main application is to Turaev's concordance group $\mathscr{VC}$ of long knots on surfaces. Boden and Nagel proved that the concordance group $\mathscr{C}$ of classical knots in $S^3$ embeds into the center of $\mathscr{VC}$. In contrast to the classical knot concordance group, we show $\mathscr{VC}$ is not abelian; answering a question posed by Turaev.
