Kanji Morimoto

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

Let K be a knot in the 3-sphere S 3 , N ( K ) the regular neighbourhood of K and E ( K ) = cl(S 3 − 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.

On the super additivity of tunnel number of knots

Abstract. In the present paper, we show a necessary and sufficient condition for knots

On the additivity of tunnel number of knots

K_1, K_2

Tunnel number, connected sum and meridional essential surfaces

in

On composite tunnel number one links

S^3

Characterization of tunnel number two knots which have the property “2 + 1 = 2”

(with some side condition) to have the super additivity of tunnel number under connected sum, i.e.,

ON TANGLE DECOMPOSITIONS OF TWISTED TORUS KNOTS

t(K_1 \# K_2) = t(K_1) + t(K_2) + 1

Characterization of composite knots with 1-bridge genus two

.

ESSENTIAL TORI IN 3-STRING FREE TANGLE DECOMPOSITIONS OF KNOTS

Abstract Let K 1 and K 2 be nontrivial knots in the 3-sphere S 3 . In this paper, we show that if the tunnel number of K 1 # K 2 is two, then either both tunnel numbers of K 1 and K 2 are one, or one of K 1 and K 2 is a 2-bridge knot and the others tunnel number is at most two.

On unknotting tunnels for knots

Abstract For given orientable closed 3-manifolds M 1 , M 2 ,…, M n and for given knots K 1 , K 2 ,…, K n in M 1 , M 2 ,…, M n , respectively, we show that if none of those 3-manifolds have lens space summands and none of those knot exteriors contain meridional essential surfaces, then the tunnel numbers of those knots do not go down under connected sum, i.e. t(K1#K2#…#Kn)⩾t(K1)+t(K2)+…+t(Kn).