The torsion of the matching group $\mathscr{C}$ of the knots of $S^3$ can be examined with the algebraic matching group $\mathscr{G}^{\mathbb{F}}$. where $\mathbb{F}$ is the field of characteristic $\chi(\mathbb{F}) \ne 2$. The group $\mathscr{G}^{\mathbb{F}}$ was defined by J. Levine and also obtained an algebraic classification when $\mathbb{F}=\mathbb{Q}$. The match group $\mathscr{C}$ is abelian, but embedded in the match group $\mathscr{VC}$ of non-Abelian virtual knots. It is unknown whether $\mathscr{VC}$ allows non-classical finite torsion. Now define a virtual algebraic match group $\mathscr{VG}^{\mathbb{F}}$ for the almost classical knots .which is analogous to $\mathscr{G}^{\mathbb{F}}$ for the homologously trivial nodes of the thick surface $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. The main result is an algebraic classification of $\mathscr{VG}^{\mathbb{F}}$. After classification, $\mathscr{G}^{\mathbb{Q}}$ is embedded in $\mathscr{VG}^{\mathbb{Q}}$ and $\mathscr{VG}^{\mathbb . {Q}}$ contains many nontrivial finite-order elements that are algebraically inconsistent with classical Seifert matrices. Generalize the Arf invariants for $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$.

Source link


Leave A Reply