Examples of tunnel number one knots which have the property ‘1 + 1 = 3’

Abstract

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.

Additivity of tunnel number for small knots

“…On the one hand, Kobayashi has exhibited knots for which the tunnel number degenerates by an arbitrarily high number under connected sum ( [6]) and on the other hand Moriah and Rubinstein and independently Morimoto, Sakuma and Yokota have exhibited knots for which the tunnel number is strictly super-additive under connected sum ( [8] and [10]). Restricting attention to connected sums of small knots circumvents some of the possibilities and many of the technical difficulties encountered in the work of Kobayashi concerning torus decompositions of manifolds and of Morimoto concerning the additivity of the tunnel numbers of knots ( [7] and [9]).…”

Heegaard Genus of the Connected Sum of M-small Knots

“…Note that in Theorem 6.3 we cannot expect every minimal genus Heegaard surface to satisfy conclusion (1), even when conclusion (2) is not satisfied. To see this, consider the following example: let K m be the knots constructed by MorimotoSakuma-Yokota in [19] and X m their exteriors. Then g(X m ) = 2 (by [19]) and g(X (1) m ) = 3 (by [10]).…”

“…To see this, consider the following example: let K m be the knots constructed by MorimotoSakuma-Yokota in [19] and X m their exteriors. Then g(X m ) = 2 (by [19]) and g(X (1) m ) = 3 (by [10]). There is an essential torus T ⊂ X m giving the decomposition X Although we do not know if K m is m-small, this example seems to indicate that replacing "there exists" with "for every" in conclusion (1) of Theorem 6.3 is not likely to be possible.…”

“…Let K m (m ∈ Z) be the knots introduced by Morimoto-Sakuma-Yokota in [19]. It is shown in [19] that for each m, g(E(K m )) = 2, K m does not admit a primitive meridian, and g(E(K m #K m )) = 4. On the other hand, in [10,Section 5] it is shown that E(K m #K m ) admits a primitive meridian.…”

Genus one knots which admit (1,1)-decompositions

“…This arc τ is called an unknotting tunnel for K. It is known that (1, 1)-knots in S 3 are tunnel number one. Morimoto, Sakuma and Yokota [8] showed that there are tunnel number one knots which do not admit (1, 1)-decompositions.…”