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

TL;DR

This paper introduces a new algebraic structure on algebraic tangles, defines algebraically alternating knots and links, and characterizes incompressible surfaces in their complements, extending classical knot theory concepts.

Contribution

It establishes a rational-like algebraic structure on algebraic tangles and characterizes incompressible surfaces in the complements of algebraically alternating knots and links.

Findings

The slope of an algebraic tangle is uniquely determined by its structure.

A new class of algebraically alternating knots and links is introduced.

Conditions for the existence of incompressible surfaces in their complements are provided.

Abstract

Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary 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. We give a necessary and sufficient condition for a closed surface to be incompressible and meridionally incompressible 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.