Bridge index for spatial θn-curves with local knots

Let b∗(Γ) be the bridge index for a spatial embedding Γ:θn→R3 of a theta n-curve (n⩾3), and Γ#K the connected sum of a spatial theta n-curve Γ and a knot K along the nth edge of Γ. In this paper, we will study interactions between the bridge index for Γ#K and those for Γ,K. Our main theorem, Theorem 0.3, shows that η(Γ,K)=b∗(Γ)+b∗(K)−b∗(Γ#K) satisfies 1⩽η(Γ,K)⩽b∗(K). This estimate is best possible. In fact, it is shown that, for any integers m,s with 1⩽m⩽s, there exists a spatial theta n-curve Γm,s satisfying η(Γs,m,K)=m for any s-bridge knot K. We also present a connected sum formula on the bridge index for spatial theta n-curves obtained from an unknotted theta n-curve In by adding local knots to all the edges of In, which generalizes Schubertʹs result in [Math. Z. 61 (1954) 245–288] on the bridge index for composite knots.