Title : Tangle sum and constructible spheres
Speaker : Koya Shimokawa (Saitama University)
Abstract :

We discuss the relation between the combinatorial properties of cell decompositions of $3$-spheres and the bridge index of knots contained in their $1$-skeletons. The main result is to solve the conjecture of Ehrenborg and Hachimori which states that for a knot $K$ in the $1$-skeleton of a constructible $3$-sphere satisfies $e(K) \ge 2b(K)$, where $e(K)$ is the number of edges $K$ consists of, and $b(K)$ is the bridge index of $K$. The key tool is a sharp inequality of the bridge index of tangles in relation with tangle sum operation, which improves the primitive rough inequality used by Eherenborg and Hachimori.

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