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Jinseok Cho

Publications and source records attributed to Jinseok Cho.

9 recordsLinked to original sources
arXiv2019

On the Hikami-Inoue conjecture

Given a braid presentation $D$ of a hyperbolic knot, Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by $D$. They show that any solution gives rise to shape parameters and thus determines a boundary-parabolic $\mathrm{PSL}(2,\mathbb{C})$-representation of the knot group. They conjecture the existence of a solution corresponding to the geometric representation. In this paper, we show that a boundary-parabolic representation $ρ$ arises from a solution if and only if the length of $D$ modulo $2$ equals the obstruction to lifting $ρ$ to a boundary-parabolic $\mathrm{SL}(2,\mathbb{C})$-representation (as an element in $\mathbb{Z}_2$). In particular, the Hikami-Inoue conjecture holds if and only if the length of $D$ is odd. This can always be achieved by adding a kink to the braid if necessary. We also explicitly construct the solution corresponding to a boundary-parabolic representation given in the Wirtinger presentation of the knot group.

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arXiv2018

Quandle theory and the optimistic limits of the representations of link groups

When a boudnary-parabolic representation of a link group to PSL(2,$\mathbb{C}$) is given, Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the shadow-coloring of certain quandle. Quandle is an algebraic system closely related with the Reidemeister moves, so their method changes quite naturally under the Reidemeister moves. In this article, we apply their method to the potential function, which was used to define the optimsitic limit, and construct a saddle point of the function. This construction works for any boundary-parabolic representation, and it shows that the octahedral triangulation is good enough to study all possible boundary-parabolic representations of the link group. Furthermore the evaluation of the potential function at the saddle point becomes the complex volume of the representation, and this saddle point changes naturally under the Reidemeister moves because it is constructed using the quandle.

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arXiv2017

Reidemeister transformations of the potential function and the solution

The potential function of the optimistic limit of the colored Jones polynomial and the construction of the solution of the hyperbolicity equations were defined in the authors' previous articles. In this article, we define the Reidemeister transformations of the potential function and the solution by the changes of them under the Reidemeister moves of the link diagram and show the explicit formulas. These two formulas enable us to see the changes of the complex volume formula under the Reidemeister moves. As an application, we can simply specify the discrete faithful representation of the link group by showing a link diagram and one geometric solution.

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arXiv2016

Conneted sum of representations of knot groups

When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume of the connected sum is the sum of each complex volumes modulo $iπ^2$ and the twisted Alexander polynomial of the connected sum is the product of each polynomials with normalization.

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arXiv2015

Optimistic limits of colored Jones polynomials and complex volumes of hyperbolic links

The optimistic limit is the mathematical formulation of the classical limit which is a physical method to expect the actual limit by using saddle point method of certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. The modified version was easier to handle and more combinatorial than the original one. On the other hand, it was known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at the certain root of unity. The optimistic limit of the colored Jones polynomial was also formulated by the author and others, but it was so complicated and needed many unnatural assumptions. In this article, we suggest a modified optimistic limit of the colored Jones polynomial, following the idea of the modified optimistic limit of the Kashaev invariant, and show that it determines the complex volume of a hyperbolic link. Furthermore, we show that this optimistic limit coincides with the optimistic limit of the Kashaev invariant modulo $4π^2$. This new version is easier to handle and more combinatorial than the old version, and has many advantages than the modified optimistic limit of the Kashaev invariant. Because of these advantages, several applications have already appeared and more are in preparation now.

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arXiv2015

Optimistic limit of the colored Jones polynomial and the existence of a solution

For the potential function of a link diagram induced by the optimistic limit of the colored Jones polynomial, we show the existence of a solution of the hyperbolicity equations by directly constructing it. This construction is based on the shadow-coloring of the conjugation quandle induced by a boundary-parabolic representation $ρ:π_1(L)\rightarrow{\rm PSL}(2,\mathbb{C})$. This gives us a very simple and combinatorial method to calculate the complex volume of $ρ$.

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arXiv2014

Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links

Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he restricted his method to knot diagrams with certain conditions. Although these restrictions are general enough for any hyperbolic knots, we have to select a good diagram of the knot to apply his theory. In this article, we suggest more combinatorial way to calculate the complex volumes of hyperbolic links using the modified optimistic limit method. This new method works for any link diagrams, and it is more intuitive, easy to handle and has natural geometric meaning.

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arXiv2011

Yokota theory, the invariant trace fields of hyperbolic knots and the Borel regulator map

For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the invariant trace field of the link. On the other hand, Yokota suggested a potential function of a hyperbolic knot, which gives the hyperbolicity equations and the complex volume of the knot. Applying the fact above to his theory, we explain that the potential function also gives all the values of the Borel regulator map and the complex volumes of the parabolic representations. Furthermore, we explain the maximum value of the imaginary parts of the complex volumes is the volume of the complete hyperbolic structure of the knot complement. Especially, if the number of the essential solutions of the hyperbolicity equations and the extension degree of the invariant trace field are the same, then the evaluation of all essential complex solutions of the hyperbolicity equations to the imaginary part of the potential function is the same with the Borel regulator map. We show these actually happens in the case of the twist knots.

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