Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A class of knots with simple $SU(2)$ representations   We call a knot in the 3-sphere $SU(2)$-simple if all representations of the
fundamental group of its complement which map a meridian to a trace-free
element in $SU(2)$ are binary dihedral. This is a generalisation of being a
2-bridge knot. Pretzel knots with bridge number $\geq 3$ are not
$SU(2)$-simple. We provide an infinite family of knots $K$ with bridge number
$\geq 3$ which are $SU(2)$-simple.
  One expects the instanton knot Floer homology $I^\natural(K)$ of a
$SU(2)$-simple knot to be as small as it can be -- of rank equal to the knot
determinant $\det(K)$. In fact, the complex underlying $I^\natural(K)$ is of
rank equal to $\det(K)$, provided a genericity assumption holds that is
reasonable to expect. Thus formally there is a resemblance to strong L-spaces
in Heegaard Floer homology. For the class of $SU(2)$-simple knots that we
introduce this formal resemblance is reflected topologically: The branched
double covers of these knots are strong L-spaces. In fact, somewhat
surprisingly, these knots are alternating. However, the Conway spheres are
hidden in any alternating diagram.
  With the methods we use, we show that an integer homology 3-sphere which is a
graph manifold always admits irreducible representations of its fundamental
group.

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The categories of this paper are: astro-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph