Let K be a knot in the 3-sphere S3, N(K) the regular neighbourhood of K and E(K) = cl(S3−N(K)) the exterior of K. The tunnel number t(K) is the minimum number of mutually disjoint arcs properly embedded in E(K) such that the complementary space of a regular neighbourhood of the arcs is a handlebody. We call the family of arcs satisfying this condition an unknotting tunnel system for K. In particular, we call it an unknotting tunnel if the system consists of a single arc.
In the present paper, we show a necessary and sufficient condition for knots K 1 , K 2 in S 3 (with some side condition) to have the super additivity of tunnel number under connected sum, i.e., t (K 1 #K 2 ) = t (K 1 ) + t (K 2 ) + 1.
In this paper, we show that there are infinitely many tunnel number two knots K such that the tunnel number of K#K' is equal to two again for any 2-bridge knot K'.
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