Research on tunnel number and unknotting tunnels of knots

设K是三维球面53中的一个纽结，那么众所周知，存在53的Heegaard分裂(Vi，V2)，使得每个手柄与K在一条平凡的圆弧中相交。我们称这种Heegaard分裂的最小亏格为K的1-桥亏格，记为g\(K)。接下来，我们可以通过连接两个给定的纽结KI，K2来考虑一个纽结，我们称它为K2，K2的连通和，记为KI#K2，然后在gI(KI#K2)=2的条件下，通过研究K1，K%的纽结类型，我们得到如下结论。(1)gI(K1)=gl(K2)=L，(2)Ki，K2，称Ki，是2-桥结，K2满足t(K2)=i，gi(K2)=2的条件，(3)Ki，Ki，K1，K%满足t(K2)=2，5-1(^2)=2的条件，属于族(7(2))。

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).