Abstract
In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have the following results. • Smallness of Montesinos tangles with length two and Kinoshita's theta curve. • Classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions.
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