arXiv 2107.096532021-07math.GT

几乎经典纽结的代数和谐序

Algebraic concordance order of almost classical knots

Micah Chrisman, Sujoy Mukherjee

备注：: 38 pages, 11 figures. v2-24 pages. Minor title change. Reorganized and shortened exposition. Some results in Section 3 and 5 removed to be placed in separate papers. Typos fixed along with other minor corrections

摘要：

在$S^3$中纽结的协调群$\mathscr{C}$中的扭转可以用代数协调群$\mathscr{G}^{\mathbb{F}}$来研究。这里$\mathbb{F}$是特征$χ(\mathbb{F}) \ne 2$的字段。$\mathscr{G}^{\mathbb{F}}$组由J. Levine定义，当$\mathbb{F}=\mathbb{Q}$。虽然一致性组$\mathscr{C}$是阿贝尔的，但它嵌入到非阿贝尔虚拟结一致性组$\mathscr{VC}$中。不知道$\mathscr{VC}$是否存在非经典有限挠。在这里，我们定义了虚拟的代数和谐组$\mathscr{VG}^{\mathbb{F}}$几乎经典的结。这是一个类似的$\mathscr{G}^{\mathbb{F}}$同调平凡结增厚表面$Σ\times [0,1]$，其中$Σ$是封闭和定向。主要结果是$\mathscr{VG}^{\mathbb{F}}$的代数分类。分类的一个结果是，$\mathscr{G}^{\mathbb{Q}}$嵌入到$\mathscr{VG}^{\mathbb{Q}}$和$\mathscr{VG}^{\mathbb{Q}}$包含许多非平凡的有限阶元素，不代数一致的任何经典的塞弗特矩阵。对于$\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$，我们给出了Arf不变量的一个推广.

英文摘要：

Torsion in the concordance group $\mathscr{C}$ of knots in $S^3$ can be studied with the algebraic concordance group $\mathscr{G}^{\mathbb{F}}$. Here $\mathbb{F}$ is a field of characteristic $χ(\mathbb{F}) \ne 2$. The group $\mathscr{G}^{\mathbb{F}}$ was defined by J. Levine, who also obtained an algebraic classification when $\mathbb{F}=\mathbb{Q}$. While the concordance group $\mathscr{C}$ is abelian, it embeds into the non-abelian virtual knot concordance group $\mathscr{VC}$. It is unknown if $\mathscr{VC}$ admits non-classical finite torsion. Here we define the virtual algebraic concordance group $\mathscr{VG}^{\mathbb{F}}$ for almost classical knots . This is an analogue of $\mathscr{G}^{\mathbb{F}}$ for homologically trivial knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. The main result is an algebraic classification of $\mathscr{VG}^{\mathbb{F}}$. A consequence of the classification is that $\mathscr{G}^{\mathbb{Q}}$ embeds into $\mathscr{VG}^{\mathbb{Q}}$ and $\mathscr{VG}^{\mathbb{Q}}$ contains many nontrivial finite-order elements that are not algebraically concordant to any classical Seifert matrix. For $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$, we give a generalization of the Arf invariant.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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