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Let $F$ be an incompressible, meridionally incompressible and notboundary-parallel surface with boundary in the complement of an algebraictangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundaryslope of $F$ is uniquely determined by $(B,T)$ and hence we can define theslope of the algebraic tangle. In addition to the Conway's tangle sum, wedefine a natural product of two tangles. The slopes and binary operation onalgebraic tangles lead an algebraic structure which is isomorphic to therational numbers.  We introduce a new knot and link class, algebraically alternating knots andlinks, roughly speaking which are constructed from alternating knots and linksby replacing some crossings with algebraic tangles. We give a necessary andsufficient condition for a closed surface to be incompressible and meridionallyincompressible in the complement of an algebraically alternating knot or link$K$, in particular we show that if $K$ is a knot, then the complement of $K$does not contain such a surface.

Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links