TY - JOUR

T1 - Non-acyclic SL2-representations of Twist Knots, -3-Dehn Surgeries, and L-functions

AU - Tange, Ryoto

AU - Tran, Anh T.

AU - Ueki, Jun

N1 - Funding Information:
This work was partially supported by Simons Foundation [#354595 to A.T.T.] and JSPS KAKENHI [grant no. JP19K14538 to J.U.]. We would like to express our sincere gratitude to L o B nard, Shinya Harada, Teruhisa Kadokami, Tomoki Mihara, Masakazu Teragaito, Yoshikazu Yamaguchi, and the anonymous referees for useful comments.
Publisher Copyright:
© 2021 The Author(s) 2019. Published by Oxford University Press. All rights reserved.

PY - 2022/7/1

Y1 - 2022/7/1

N2 - We study irreducible SL2-representations of twist knots. We first determine all nonacyclic SL2(C)-representations, which turn out to lie on a line denoted as x = y in R2. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on L-functions, that is, the orders of the knot modules associated to the universal deformations. 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 nonacyclic representations over a finite field with characteristic p < 2, to concretely determine all non-trivial L-functions L? of the universal deformations over complete discrete valuation rings. We show among other things that L = kn(x)2 holds for a certain series kn(x) of polynomials.

AB - We study irreducible SL2-representations of twist knots. We first determine all nonacyclic SL2(C)-representations, which turn out to lie on a line denoted as x = y in R2. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on L-functions, that is, the orders of the knot modules associated to the universal deformations. 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 nonacyclic representations over a finite field with characteristic p < 2, to concretely determine all non-trivial L-functions L? of the universal deformations over complete discrete valuation rings. We show among other things that L = kn(x)2 holds for a certain series kn(x) of polynomials.

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U2 - 10.1093/imrn/rnab034

DO - 10.1093/imrn/rnab034

M3 - Article

AN - SCOPUS:85135697621

SN - 1073-7928

VL - 2022

SP - 11690

EP - 11731

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 15

ER -