Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tangle Equations, the Jones conjecture, and quantum continued fractions   We study systems of $2$-tangle equations $$ N(X+T_1)=L_1,\quad
N(X+T_2)=L_2,$$ which play an important role in the analysis of enzyme actions
on DNA strands.
  We show the benefits of considering such systems in the context of framed
tangles and, in particular, we conjecture that in this setting each such system
has at most one solution $X.$ We prove a version of this statement for rational
tangles.
  More importantly, %we establish a connection between systems of tangle
equations and the Jones conjecture. Specifically, we show that the Jones
conjecture implies that if a system of tangle equations has a rational solution
then that solution is unique among all tangles. This result potentially opens a
door to a purely topological line of attack on the Jones conjecture.
  Additionally, we relate systems of tangle equations to the Cosmetic Surgery
Conjecture.
  Furthermore, we establish a number of properties of the Kauffman bracket
$([T]_0,[T]_\infty)$ of $2$-tangles $T$ for the purpose of the proofs of the
above results, which are of their own independent interest. In particular, we
show that the Kauffman bracket ratio $Q(T)=[T]_\infty/[T]_0$ quantizes
continued fraction expansions of rationals. Furthermore, we prove that for
algebraic tangles $Q(T)$ determines the slope of incompressible surfaces in
$D^3\smallsetminus T$.

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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: hep-ph