High distance tangles and tunnel number of knots

Abstract

We show that for any integers ti ⩾ 0 (i = 1, 2) and n ⩾ 2, there is a knot K in the 3-sphere with an n-tangle decomposition K = T1∪T2 such that tnl(Ti) = ti (i = 1, 2) and that tnl(K) = tnl(T1) + tnl(T2) + 2n − 1, where tnl(⋅) is the tunnel number. This contains an affirmative answer to an unsolved problem asked by Morimoto.