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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.
Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental grou
We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instant
A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic
Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5-chain link. In this article we enumerate all hyperbolic knots in S^3 obtained by surgery on the minimally twisted 5 chain link that realize the m
We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky theory for