Additivity of circular width (Q2885411)

\textit{M. Scharlemann} and \textit{A. Thompson} [Contemp. Math. 164, 231--238 (1994; Zbl 0818.57013)] introduced the concept of thin position for a compact \(3\)-manifold. Thin position is a generalized Heegaard splitting which minimizes a certain integer-based complexity, called the width of the manifold, over all the possible interval-valued Morse functions on the manifold. In [Algebr. Geom. Topol. 9, No. 1, 429--454 (2009; Zbl 1171.57005)], the second author of this paper defined circular thin position with circular width which generalizes thin position by taking a circle-valued Morse function in the case that the manifold is the exterior \(E(K)\) of a knot \(K\) in the \(3\)-sphere. Further, given two knots \(K_1\) and \(K_2\), an upper bound for the circular width of \(E(K_1 \sharp K_2)\) was obtained, which depends on the circular widths of the original knot exteriors. Namely, \(\text{cw}(E(K_1 \sharp K_2)) \leq \text{cw}(E(K_1)) \sharp \text{cw}(E(K_2))\).NEWLINENEWLINEIn this paper, the authors show that the equality of the above formula holds if (1) \(K_1\) and \(K_2\) are fibered, (2) \(K_1\) is fibered and \(K_2\) is not fibered or (3) \(K_1\) and \(K_2\) are not fibered, but \(E(K_1)\) and \(E(K_2)\) have circular thin positions containing minimal genus Seifert surfaces as a thin level.