arXiv subjects

Kyeonghee Jo

Publications and source records attributed to Kyeonghee Jo.

8 recordsLinked to original sources

arXiv2026

Symplectic quandle Method and $SL(2,\mathbb C)$-representations of 2-bridge Knots

In this paper, we extend the symplectic quandle method, previously employed in our study of parabolic representations of knot groups, to investigate the general $SL(2,\mathbb{C})$-representations of 2-bridge ``kmot" groups. We introduce a `generalized symplectic quandle structure' corresponding to ($\mathcal{D}_M$, conjugation) for each $M\in\mathbb C\setminus \{0,1,-1\}$, where $\mathcal{D}_M=\{A\in SL(2,\mathbb{C})\mid tr(A)= M+M^{-1} \}$. By converting the system of conjugation quandle equations to that of generalized symplectic quandle equations, we obtain a simpler expression for the 2-variable Riley polynomial and derive some recursive formulas for Riley polynomials and Alexander polynomials. This approach enables us to effectively compute the A-polynomials, allowing us to obtain numerous previously unknown A-polynomials within minutes using Mathematica.

math.GT↗

arXiv2024

Non-freeness of parabolic two-generator groups

A complex number $λ$ is said to be non-free if the subgroup of $SL(2,\bc)$ generated by $$X=\begin{pmatrix} 1& 1\\ 0 & 1 \end{pmatrix} \,\, \text{and}\,\,\,Y_λ=\begin{pmatrix} 1& 0\\ λ& 1 \end{pmatrix}$$ is not a free group of rank 2. In this case the number $λ$ is called a relation number, and it has been a long standing problem to determine the relation numbers. In this paper, we characterize the relation numbers by establishing the equivalence between $λ$ being a relation number and $u:=\sqrt{- λ}$ being a root of a `generalized Chebyshev polynomial'. The generalized Chebyshev polynomials of degree $k$ are given by a sequence of $k$ integers $(n_1, n_2,\cdots, n_k)$ using the usual recursive formula, and thereby can be studied systematically using continuants and continued fractions. Such formulation, then, enables us to prove that, the question whether a given number $λ$ is a relation number of $u$-degree $k$ can be answered by checking only finitely many generalized Chebyshev polynomials. Based on these theorems, we design an algorithm deciding any given number is a relation number with minimal degree $k$. With its computer implementation we provide a few sample examples, with a particular emphasis on the well known conjecture that every rational number in the interval $(-4, 4)$ is a relation number.

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arXiv2022

Continuant, Chebyshev polynomials, and Riley polynomials

In the previous paper, we showed that the Riley polynomial $\mathcal{R}_K(λ)$ of each 2-bridge knot $K$ is split into $\mathcal{R}_K(-u^2)=\pm g(u)g(-u)$, for some integral coefficient polynomial $g(u)\in \mathbb Z[u]$. In this paper, we study this splitting property of the Riley polynomial. We show that the Riley polynomial can be expressed by `$ε$-Chebyshev polynomials', which is a generalization of Chebyshev polynomials containing the information of $ε_i$-sequence $(ε_i=(-1)^{[i\fracβα]})$ of the 2-bridge knot $K=S(α,β)$, and then we give an explicit formula for the splitting polynomial $g(u)$ also as $ε$-Chebyshev polynomials. As applications, we find a sufficient condition for the irreducibility of the Riley polynomials and show the unimodal property of the symmetrized Riley polynomial.

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arXiv2020

Symplectic quandles and parabolic representations of 2-bridge Knots and Links

In this paper we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial $P_K(u)$ for each 2-bridge link $K$, and each zero of this polynomial gives a set of arc coloring vectors on the diagram of $K$ satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of $K$. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

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arXiv2019

On the Markus conjecture in convex case

In this paper, we show that any convex affine domain with a nonempty limit sets on the boundary under the action of the identity component of the automorphism group cannot cover a compact affine manifold with a parallel volume, which is a positive answer to the Markus conjecture for convex case. Consequently, we show that the Markus conjecture is true for convex affine manifolds of dimension $\leq 5$.

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arXiv2010

SO(n)\SO_0(n,1) has Positive Curvatures

The Lie group SO_0(n, 1) has the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left. The geometry of the quotient space of the homogeneous submersion SO_0(n, 1) -> SO(n)\SO_0(n, 1) is investigated. The space is expressed as a warped product. Its group of isometries and sectional curvatures are calculated.

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arXiv2003

Quasi-homogeneous domains and convex affine manifolds

In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least $C^1$ boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional paraboloid is the only strictly convex quasi-homogeneous affine domain in $\mathbb R^n$ up to affine equivalence. Furthermore we prove that if a strictly convex quasi-homogeneous projective domain is $C^α$ on an open subset of its boundary, then it is $C^α$ everywhere. Using this fact and the properties of asymptotic cones we find all possible shapes for developing images of compact convex affine manifolds with dimension $\leq 4$.

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arXiv2002

Invariant Measure and the Euler Characteristic of Projectively Flat Manifolds

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP^n invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that the Chern's conjecture is true for a closed affinely flat manifold whose holonomy group action permits an invariant probability Borel measure on RP^n; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.

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