Rational Structure on Algebraic Tangles and Closed Incompressible Surfaces in Algebraically Alternating Knots and Links

Let F be an incompressible, meridionally incompressible and not boundary-parallel surface in the complement of an algebraic tangle (B, T ). Then F separates the strings of T in B and the boundary slope of F is uniquely determined by (B, T ) and hence we can define the slope of the algebraic tangle. In addition to the Conway’s tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead an algebraic structure which is isomorphic to the rational numbers. We introduce a new knot and link class, algebraically alternating knots and links, roughly speaking which are constructed from alternating knots and links by replacing some crossings with algebraic tangles. Let F be a closed incompressible and meridionally incompressible surface in the complement of an algebraically alternating knot or link K. Then F separates the components of K, in particular if K is a knot, then the complement of K does not contain such a surface.