Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rational structure on algebraic tangles and closed incompressible
  surfaces in the complements of algebraically alternating knots and links   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.

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The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess