标题： genus one 三维球面中的双曲纽结的 Kakimizu 复形 显示英文标题

标题： The Kakimizu complex for genus one hyperbolic knots in the 3-sphere

摘要： The Kakimizu complex $MS(K)$ for a knot $K\subset\mathbb{S}^3$ is the simplicial complex with vertices the isotopy classes of minimal genus Seifert surfaces in the exterior of $K$ and simplices any set of vertices with mutually disjoint representative surfaces. In this paper we determine the structure of the Kakimizu complex $MS(K)$ of genus one hyperbolic knots $K\subset\mathbb{S}^3$. 与高 genus 双曲扭结的情况相反，已知 $MS(K)$ 的维度 $d$ 被普遍地限制为 $4$，我们证明当 $d=0,4$ 时， $MS(K)$ 由一个单个的 $d$-单形组成，否则由最多两个相交于公共 $(d-1)$-面的 $d$-单形组成。 对于情况$1\leq d\leq 3$，我们还构造了此类纽结的无限多个例子，其中$MS(K)$由两个$d$-单形组成。 显示英文摘要

摘要： The Kakimizu complex $MS(K)$ for a knot $K\subset\mathbb{S}^3$ is the simplicial complex with vertices the isotopy classes of minimal genus Seifert surfaces in the exterior of $K$ and simplices any set of vertices with mutually disjoint representative surfaces. In this paper we determine the structure of the Kakimizu complex $MS(K)$ of genus one hyperbolic knots $K\subset\mathbb{S}^3$. In contrast with the case of hyperbolic knots of higher genus, it is known that the dimension $d$ of $MS(K)$ is universally bounded by $4$, and we show that $MS(K)$ consists of a single $d$-simplex for $d=0,4$ and otherwise of at most two $d$-simplices which intersect in a common $(d-1)$-face. For the cases $1\leq d\leq 3$ we also construct infinitely many examples of such knots where $MS(K)$ consists of two $d$-simplices.