A chiral polytope with Schläfli symbol \(\{p_1, \ldots , p_{n-1}\}\) has at least \(2p_1 \cdots p_{n-1}\) flags, and it is called tight if the number of flags meets this lower bound. The Schläfli symbols of tight chiral polyhedra were classified in an earlier paper, and another paper proved that there are no tight chiral n-polytopes with \(n \ge 6\). Here we prove that there are no tight chiral 5-polytopes, describe 11 families of tight chiral 4-polytopes, and show that every tight chiral 4-polytope covers a polytope from one of those families.

中文翻译：

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紧密手性多表位

与施莱夫利符号的手性多面体\（\ {P_1，\ ldots，P_ {N-1} \} \）具有至少\（2p_1 \ cdots P_ {N-1} \）标志，它被称为紧如果标志数满足此下限。较紧密的手性多面体的Schläfli符号已在较早的论文中进行了分类，另一篇论文证明了不存在带\（n \ ge 6 \）的紧密手性n-多义位。在这里，我们证明没有紧密的手性5-多聚体，描述了11个紧密的手性4-多聚体家族，并表明每个紧密的手性4-多聚体都覆盖了其中一个家族的多聚体。