一些不存在SU(2)-阿贝尔手术的纽结
Some knots with no SU(2)-abelian surgeries
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中文总结 AI 辅助
本文构造非SU(2)-阿贝尔纽结例子，证明非平凡SU(2)-干净纽结的连通和非SU(2)-阿贝尔，还得出交叉数≤9的非环面纽结除9₄₇、9₄₉外均非SU(2)-阿贝尔的结论。
中文摘要 AI 辅助
若S³中的纽结沿其进行的所有非平凡手术得到的3维流形的基本群都存在不可约SU(2)表示，则称该纽结为非SU(2)-阿贝尔纽结。本文构造了S³中此类非SU(2)-阿贝尔纽结的例子。若整数r-手术的基本群π₁(K(r))没有SU(2)不可约表示时，纽结K的亚历山大多项式在所有r次单位根处均非零，则称K为SU(2)-干净纽结。本文证明，若K是非平凡SU(2)-干净纽结，则其连通和K#K永远不是SU(2)-阿贝尔纽结。最后，结合SU(2)-丰富纽结的已知结果与纽结群间的经典满同态，本文证明，交叉数不超过9的所有非环面纽结，除9₄₇和9₄₉这最多两个例外，均非SU(2)-阿贝尔纽结。
英文摘要
A knot in $S^3$ is said to be \emph{not} $SU(2)$-abelian knot, if every non-trivial surgery along it yields a $3$-manifold whose fundamental group admits an irreducible $SU(2)$-representation. We provide examples of knots in $S^3$ that are not $SU(2)$-abelian. A knot $K$ is said to be $SU(2)$-\emph{clean} if whenever the fundamental group of the integer $r$-surgery $\fund{K(r)}$ has no $SU(2)$-irreducible representation, then the Alexander polynomial of $K$ does not vanish at any $r$-th root of unity. We show that if $K$ is a non-trivial $SU(2)$-clean knot, the connected sum $K\#K$ is never $SU(2)$-abelian. Finally, combining known results on $SU(2)$-abundant knots and a classical epimorphism between knot groups, we show that \emph{every} non-torus knot with at most $9$ crossings is not $SU(2)$-abelian, with at most the two exceptions of $9_{47}$ and $9_{49}$.
