Boundary Stabilizations Of Heegaard Splittings And Tunnel Numbers Of Cable Knots

Given a Heegaard splitting of a compact 3-manifold,Hempel defined the concept of distance to describe its complexity.Firstly,for two essential simple closed curves in a closed surface of genus at least 2,Hempel defined the distance between them to be one more than the minimal number of essential simple closed curves connecting them which are disjoint adjacently.Then Hempel defined the distance of a Heegaard splitting(1?(2 to be the minimal number of the distance on the the Heegaard surfacebetween all simple closed curves bounding essential disks in(1 and all simple closed curves bounding essential disks in(2.Here we assume the genus ofis at least 2.For a Heegaard splitting with distance at least 2,we can define the concept of locally large.A distance at least 2 Heegaard splitting(1?(2 is called locally large,if in a set of essential simple closed curves onwhich realize the distance,for any three adjacent closed curves,the distance between those projecting the two curves on both ends to the subsurface which is obtained by cutting the surfacealong the middle curve is sufficiently large.The first research problem of this dissertation is when the boundary stabiliza-tion of a Heegaard splitting of a compact 3-manifold is unstabilized.Zou-Guo-Qiu proved that the boundary stabilization of a distance at least 6 Heegaard splitting is unstabilized.In fact,their lower bound can be improved to 5.Since there is a distance 2 Heegaard splitting whose boundary stabilization is stabilized,they conjectured that 3 is the sharp lower bound.Here we give an affirmative answer to their conjecture in the case of adding a condition:The boundary stabilization of a distance at least 3,locally large Heegaard splitting is unstabilized(Theorem3.1).There are some results guaranteeing the existence of locally large Heegaard splittings.Qiu-Zou-Guo constructed infinitely many distance at least 4 Heegaard splittings.In fact,their examples are locally large Heegaard splittings.Zhang-Qiu-Zou constructed infinitely many similar examples for the distance 2 and 3case.Hence there are infinitely many distance at least 3,locally large Heegaard splittings.Another research problem of this dissertation is the relationship between tunnel numbers of a cable knot and its companion knot.The tunnel number of a knot in the three sphere~3is one less than the Heegaard genus of its complement in~3.By a simple observation,we know that the tunnel number of a cable knot is no more than the tunnel number of its companion knot plus 1.When the tunnel number of the companion knot is 1,Moriah proved that the tunnel number of the cable knot can reach this upper bound.Here we first prove that the tunnel number of a cable knot is no less than the tunnel number of its companion knot(Theorem 4.1).Then the tunnel number of a cable knot is equal to the tunnel number of its companion knot or plus 1.Next we give two sufficient conditions for the tunnel number of a cable knot is equal to the tunnel number of its companion knot plus 1:One requires that the companion knot is high distance;the other requires that the gluing map is sufficiently complicated in the construction of a cable knot,i.e.,the gluing slope is far away from a finite set which consists of boundary curves of properly embedded incompressible surfaces in the curve complex of a torus(Theorem 4.2).Finally,we give some applications.