On the degeneration ratio of tunnel numbers and free tangle decompositions of knots

Let K be a knot in the 3–sphere S3 , t(K) the tunnel number of K and K1#K2 the connected sum of two knots K1 and K2 , where t(K) is the minimal genus −1 among all Heegaard splittings which contain K as a core of a handle. Concerning the relationship between t(K1)+ t(K2) and t(K1#K2), we showed in Morimoto [2] that there are infinitely many tunnel number two knots K such that t(K#K′) is two again for any 2–bridge knots K′ . These are the first examples whose tunnel numbers go down under connected sum, ie, “2+1 = 2". Subsequently, Kobayashi showed in Kobayashi [1], by taking connected sum of those knots, that there are infinitely many pairs of knots (K1,K2) such that t(K1#K2) < t(K1) + t(K2)− n for any integer n > 0. This shows that tunnel numbers of knots have arbitrarily high degeneration.