We study several properties of the completed group ring ℤ [ [ t ℤ ] ] and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots J and K are isomorphic, then their Alexander polynomials Δ J ( t ) and Δ K ( t ) coincide.

https://msp.org/agt/2018/18-5/agt-v18-n5-p14-s.pdf

The profinite completions of knot groups determine the Alexander polynomials

We study several properties of the completed group ring $\widehat{\mathbb{Z}}[[t^{\widehat{\mathbb{Z}}}]]$ and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots $J$ and $K$ are isomorphic, then their Alexander polynomials $\Delta_J(t)$ and $\Delta_K(t)$ coincide.

We study the twisted knot module for the universal deformation of an SL2-representation of a knot group, and introduce an associated L-function, which may be seen as an analogue of the algebraic p-adic Lfunction associated to the Selmer module for the universal deformation of a Galois representation. We then verify affirmatively Mazur’s problem on the order of a zero of the L-function by some examples for 2-bridge knots.

/pdf/on-certain-l-functions-for-deformations-of-knot-group-3p5jlvbosv.pdf

On certain l-functions for deformations of knot group representations

Idèlic class field theory for 3-manifolds and very admissible links

\n We study irreducible $\\mathop{\\textrm{SL}}\\nolimits _2$-representations of twist knots. We first determine all non-acyclic $\\mathop{\\textrm{SL}}\\nolimits _2({\\mathbb{C}})$-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 $L$-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 non-acyclic representations $\\overline{\\rho }$ over a finite field with characteristic $p>2$, to concretely determine all non-trivial $L$-functions $L_{{\\boldsymbol{\\rho }}}$ of the universal deformations over complete discrete valuation rings. We show among other things that $L_{{\\boldsymbol{\\rho }}}$  $\\dot{=}$  $k_n(x)^2$ holds for a certain series $k_n(x)$ of polynomials.

http://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnab034/37098672/rnab034.pdf

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

Based on the analogies between knot theory and number theory, we study a deformation theory for ${\rm SL}_2$-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-${\rm SL}_2$-representations, we prove the existence of the universal deformation of a given ${\rm SL}_2$-representation of a finitely generated group $\Pi$ over a perfect field $k$ whose characteristic is not 2. We then show its connection with the character scheme for ${\rm SL}_2$-representations of $\Pi$ when $k$ is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures.

On the universal deformations for SL2-representations of knot groups

Based on the analogies between knot theory and number theory, we study a deformation theory for SL_2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL_2-representations, we prove the existence of the universal deformation of a given SL_2-representation of a finitely generated group Pi over a field whose characteristic is not 2. We then show its connection with the character scheme for SL_2-representations of Pi when k is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures.

Yu Takakura

Papers

On the universal deformations for SL2-representations of knot groups

On the universal deformations for SL_2-representations of knot groups