Seifert流形的SO（3）不变量及其代数可积性
SO(3) invariants of Seifert manifolds and their algebraic integrality
摘要：
对于Seifert流形，计算所有$r$奇数$\geq 3$的$M=X({p_1}/_{\f{q_1}},{p_2}/_{\f{q_2}}, ...,{p_n}/_ {\f{q_n}}), τ^{'}_r(M)$。若$r$至少与$p_k$的$n-2$互质（例如当$M$是Poincare同调球面时），则证明了$(\sqrt {\dfrac{4}{r}}\sin \dfracπ{r})^ντ^{'}_r(M)$是r阶分圆域上的代数整数，其中$ν$是$M$的第一个Betti数.对于框架为0的三叶结环面丛，如果$3\mid\llap /r$，则得到了$X_{tref}(0)=X(-2/_{\f{1}},3/_{\f{1}},6/_{\f{1}}), τ^{'}_r$的简单形式，这在某种意义上表明，将Ohtsuki不变量推广到非有理同调球面的三维流形是不可能的.
英文摘要：
For Seifert manifold $M=X({p_1}/_{\f{q_1}},{p_2}/_{\f{q_2}}, ...,{p_n}/_ {\f{q_n}}), τ^{'}_r(M)$ is calculated for all $r$ odd $\geq 3$. If $r$ is coprime to at least $n-2$ of $p_k$ (e.g. when $M$ is the Poincare homology sphere), it is proved that $(\sqrt {\dfrac{4}{r}}\sin \dfracπ{r})^ντ^{'}_r(M)$ is an algebraic integer in the r-th cyclotomic field, where $ν$ is the first Betti number of $M$. For the torus bundle obtained from trefoil knot with framing 0, i.e. $X_{tref}(0)=X(-2/_{\f{1}},3/_{\f{1}},6/_{\f{1}}), τ^{'}_r$ is obtained in a simple form if $3\mid\llap /r$, which shows in some sense that it is impossible to generalize Ohtsuki's invariant to 3-manifolds being not rational homology spheres.
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