Handle Additions And Incompressible Surfaces

The 3-manifold theory is one of the important branchs of the low dimension topology. At present, the main methods of studying the 3-manifolds include algebraic method, geometrical method and combinatorial method. In the present paper, we mainly use the combinatorial method to discuss some questions.The studying of Heegaard splitting, Dehn surgery, handle additions and incompressible surfaces are the important subjects in combinatorial topology of the 3-manifolds. And both Heegaard splitting and Dehn surgery of the 3-manifolds can be viewed as handle additions. In the present paper, we discuss the reducible and (?)-reducible handle additions on a hyperbolic 3-manifold. And we also discuss the existence of arbitrarily high genus incompressible surfaces in the knots complements.In 1984, C. Gordon and R. Litherland set up the system of labeling graph theory, which is a very useful tool in the studying of Dehn surgery, and then C. Gordon and J. Luecke develop the system. After that, almost all upper bound of the degenerating curves in each case about Dehn surgery has been well estimated. Today, people pay more attention to the general handle additions on hyperbolic 3-manifolds, especially, on separating degenerating handle additions. M. Scharlemann and Y-Q. Wu proved the theorem: Suppose M is hyperbolic and F is a boundary component of M with genus g≥2. Letα,βbe separating curves on F. If M[α] and M[β] are nonhyperbolic, thenΔ(α,β)≤14. In the present paper, we find and set up the weakly parity rule of the handle additions, and at the same time, we extend the concept of the Scharlemann circle. Then, many known results about the labeling graph theory in Dehn surgery can be applied in the research of the handle additions. Using these methods, we can obtain finer conclusions. Suppose M is hyperbolic and F is a boundary component of M with genus g = 2. Then there is at most one separating (?)-reducible handle addition on F. Together with the result of Scharlemann and Wu's and the result of Qiu and Zhang's, we have: Suppose M is a hyperbolic and F is a boundary component of M with genus g = 2, then there is at most one separating curve 7 on F such that M[γ] is reducible or (?)-reducible.The study of the incompressible surfaces is one of the important problems of the 3-manifold theory. It is known by several authors that there are 3-manifolds each of which admits arbitrarily high genus incompressible surfaces. As a sequel of these results, in this paper, we show that if k₁ is a knot which admits a 2-string essential free tangle decompo- sition, then the exterior of the connected sum k₁#k₂ for any non-trivial knot k₂ admits a closed incompressible surface of genus n for each positive integer n.