Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-acyclic ${\rm SL}_2$-representations of twist knots, $-3$-Dehn
  surgeries, and $L$-functions   We study irreducible ${\rm SL}_2$-representations of twist knots. We first
determine all non-acyclic ${\rm SL}_2(\mathbb{C})$-representations, which turn
out to lie on a line denoted as $x=y$ in $\mathbb{R}^2$. Our main tools are
character variety, Reidemeister torsion, and Chebyshev polynomials. We also
verify a certain common tangent property, which yields a result on the
$L$-functions of universal deformations, that is, the orders of the associated
knot modules. Secondly, we prove that a representation is on the line $x=y$ if
and only if it factors through the $(-3)$-Dehn surgery, and is non-acyclic if
and only if the image of a certain element is of order 3. Finally, we study
absolutely irreducible non-acyclic representations $\overline{\rho}$ over a
finite field with characteristic $p>2$ to concretely determine all non-trivial
$L$-functions $L_{\rho}$ of the universal deformations over a CDVR. We show
among other things that $L_{\rho}$ $\dot{=}$ $k_n(x)^2$ holds for a certain
series $k_n(x)$ of polynomials.

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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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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: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin