TANGLE SUM AND CONSTRUCTIBLE SPHERES

In this paper 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)≥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. We also present an application of our new tangle sum inequality to improve Armentrout's result on the relation between shellability of cell decompositions of 3-spheres and the bridge index of knots in a general position to their 2-skeletons.

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