Journal of Knot Theory and Its Ramifications, 13 (2004), 373-383.
## Tangle sum and constructible spheres.

by Masahiro Hachimori and Koya Shimokawa

## Abstract

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 $e(K) \ge 2b(K)$, 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.
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.