Research on tunnel number and unknotting tunnels of knots

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Let K be a knot in the 3-dimensional sphere 53.Then it is well known that there is a Heegaard splitting (Vi, V2) of 53 such that each handlebody intersects K in a single trivial arc. We call the minimal genus of such Heegaard splittings the 1-bridge genus of K and denote it by g\(K). Next we can consider a knot by connecting two given knots Ki, K2.We call it the connected sum of K\, K2 and denote it by Ki#K2.Then by invesitigating the knot types of Ki, K% under the condition that gi(Ki#K2) = 2, we get the following. (1) gi(Kl) =gl(K2) = l, (2) One of Ki, K2, say KI, is a 2-bridge knot and K2 satisfies the condition that t(K2) = I andgi(K2) = 2, (3) One of K\, Ki, say K\, is a 2-bridge knot and K% satisfies the condition that t(K2) = 2, 5-1(^2) = 2 and belongs to the family (7(2).