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Seokbeom Yoon

Publications and source records attributed to Seokbeom Yoon.

At least 19 recordsLinked to original sources
arXiv2026

Computations of parabolic character schemes of knots

We compute parabolic $\mathrm{SL}_2(\mathbb{C})$-character schemes of knots using the parabolic quandle. To this end, we introduce sign-refined arc-colorings and show that their sign data encode the obstruction classes of the induced parabolic representations. We also establish a correspondence between the schemes defined by sign-refined arc-colorings and the parabolic character scheme. This yields a practical diagrammatic method for computing complete lists of parabolic characters, together with their multiplicities and obstruction classes. Using this method, we verify a conjecture of B\'{e}nard and Detcherry for all small knots with at most $12$ crossings.

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arXiv2026

Knots with large character varieties

We study knots whose $\mathrm{SL}_2(\mathbb{C})$-character varieties have a component of dimension greater than one. We call such knots $\mathcal{X}$-large and introduce two diagrammatic constructions that produce $\mathcal{X}$-large knots. The first construction uses split link diagrams and rational tangle replacements, providing a topological explanation for most $\mathcal{X}$-large knots observed in knot tables. The second construction is based on braids and orientation-reversing involutions, and is motivated by a detailed analysis of the knot $10_{123}$, also known as the Turk's head knot $Th(3,5)$. In particular, this approach applies to Turk's head knots $Th(p,q)$ with $p$ and $q$ odd, leading us to conjecture that all such knots are $\mathcal{X}$-large. In doing so, we also present a non-orientable analogue of Thurston's theorem giving a lower bound on the dimension of character varieties of non-orientable 3-manifolds.

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arXiv2025

Reidemeister torsion of two-bridge knots and signatures of TQFT

We establish an explicit relation between the adjoint Reidemeister torsion of the two-bridge knot $K(p,q)$ at any parabolic representation and the Frobenius algebra governing the signatures of SU$_2$-TQFT vector spaces at the root $\zeta=\exp(i\pi q/p)$. As applications, (a) we prove that the inverse sum of torsions is constant (i.e., independent of $p$ and $q$); and (b) we show that along sequences of roots of the form $\zeta_n = \exp\left(i\pi\tfrac{a+bn}{c+dn}\right)$, the signatures have the same asymptotic behavior as the Verlinde formula.

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arXiv2024

1-loop equals torsion for two-bridge knots

Motivated by the conjectured asymptotics of the Kashaev invariant, Dimofte and the first author introduced a power series associated to a suitable ideal triangulation of a cusped hyperbolic 3-manifold, proved that its constant (1-loop) term is a topological invariant and conjectured that it equals to the adjoint Reidemeister torsion. We prove this conjecture for hyperbolic 2-bridge knots by combining the work of Ohtsuki--Takata with an explicit computation.

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arXiv2024

The (twisted/$L^2$)-Alexander polynomial of ideally triangulated 3-manifolds

We establish a connection between the Alexander polynomial of a knot and its twisted and $L^2$-versions with the triangulations that appear in 3-dimensional hyperbolic geometry. Specifically, we introduce twisted Neumann--Zagier matrices of ordered ideal triangulations and use them to provide formulas for the Alexander polynomial and its variants, the twisted Alexander polynomial and the $L^2$-Alexander torsion.

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arXiv2023

Octahedral developing of knot complement II: Ptolemy coordinates and applications

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston's gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a $(\mathrm{PSL}(2,\mathbb{C}),P)$-representation of the knot group to a $(\mathrm{SL}(2,\mathbb{C}),P)$-representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group.

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arXiv2023

Asymptotically multiplicative quantum invariants

The Euler characteristic and the volume are two best-known multiplicative invariants of manifolds under finite covers. On the other hand, quantum invariants of 3-manifolds are not multiplicative. We show that a perturbative power series, introduced by Dimofte and the first author and shown to be a topological invariant of cusped hyperbolic 3-manifolds by Storzer--Wheeler and the first author, and conjectured to agree with the asymptotics of the Kashaev invariant to all orders in perturbation theory, is asymptotically multiplicative under cyclic covers. Moreover, its coefficients are determined by polynomials constructed out of twisted Neumann--Zagier data. This gives a new $t$-deformation of the perturbative quantum invariants, different than the $x$-deformation obtained by deforming the geometric representation. We illustrate our results with several hyperbolic knots.

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arXiv2023

1-loop equals torsion for fibered 3-manifolds

In earlier work of two of the authors, two 1-loop polynomial invariants of cusped 3-manifolds were constructed using combinatorial data of ideal triangulations, and conjectured to be equal to the $\mathbb{C}^2$ and the $\mathbb{C}^3$-torsion polynomials. Here, we prove this conjecture for layered triangulations of fibered 3-manifolds with toroidal boundary, and we illustrate our theorems with exact computations of the 1-loop and the torsion polynomials. As further evidence for the conjecture, we confirm it for more than 6,600 nonfibered manifolds, and use this data to explore the extent to which the $\mathbb{C}^2$-torsion determines the Thurston norm.

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arXiv2023

Super-representations of 3-manifolds and torsion polynomials

Torsion polynomials connect the genus of a hyperbolic knot (a topological invariant) with the discrete faithful representation (a geometric invariant). Using a new combinatorial structure of an ideal triangulation of a 3-manifold that involves edges as well as faces, we associate a polynomial to a cusped hyperbolic manifold that conjecturally agrees with the $\BC^2$-torsion polynomial, which conjecturally detects the genus of the knot. The new combinatorics is motivated by super-geometry in dimension 3, and more precisely by super-Ptolemy assignments of ideally triangulated 3-manifolds and their $\mathrm{OSp}_{2|1}(\BC)$-representations. Extended section 4, and added superalgebras with a single odd generator.

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arXiv2022

Parabolic representations and generalized Riley polynomials

We generalize R. Riley's study about parabolic representations of two bridge knot groups to the general knots in $S^3$. We utilize the parabolic quandle method for general knot diagrams and adopt symplectic quandle for better investigation, which gives such representations and their complex volumes explicitly. For any knot diagram with a specified crossing $c$, we define a generalized Riley polynomial $R_c(y) \in \mathbb{Q}[y]$ whose roots correspond to the conjugacy classes of parabolic representations of the knot group. The sign-type of parabolic quandle is newly introduced and we obtain a formula for the obstruction class to lift to a boundary unipotent $\text{SL}_2 \mathbb{C}$-representation. Moreover, we define another polynomial $g_c(u)\in\mathbb{Q}[u]$, called $u$-polynomial, and prove that $R_c(u^2)=\pm g_c(u)g_c(-u)$. Based on this result, we introduce and investigate Riley field and $u$-field which are closely related to the invariant trace field. This method eventually leads to the complete classification of parabolic representations of knot groups along with their complex volumes and cusp shapes up to 12 crossings.

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arXiv2021

The twisted 1-loop invariant and the Jacobian of Ptolemy coordinates

We present an alternative definition of the twisted 1-loop invariant in terms of the Jacobian of Ptolemy coordinates. As an application, we prove that the twisted 1-loop invariant is equal to the adjoint twisted Alexander polynomial for all hyperbolic once-punctured torus bundles. This implies that the 1-loop conjecture proposed by Dimofte and Garoufalidis holds for all hyperbolic once-punctured torus bundles.

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arXiv2021

Twisted Neumann--Zagier matrices

The Neumann--Zagier matrices of an ideal triangulation are integer matrices with symplectic properties whose entries encode the number of tetrahedra that wind around each edge of the triangulation. They can be used as input data for the construction of a number of quantum invariants that include the loop invariants, the 3D-index and state-integrals. We define a twisted version of Neumann--Zagier matrices, describe their symplectic properties, and show how to compute them from the combinatorics of an ideal triangulation. As a sample application, we use them to define a twisted version of the 1-loop invariant (a topological invariant) which determines the 1-loop invariant of the cyclic covers of a hyperbolic knot complement, and conjecturally equals to the adjoint twisted Alexander polynomial.

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arXiv2021

The adjoint Reidemeister torsion for the connected sum of knots

Let $K$ be the connected sum of knots $K_1,\ldots,K_n$. It is known that the $\mathrm{SL}_2(\mathbb{C})$-character variety of the knot exterior of $K$ has a component of dimension $\geq 2$ as the connected sum admits a so-called bending. We show that there is a natural way to define the adjoint Reidemeister torsion for such a high-dimensional component and prove that it is locally constant on a subset of the character variety where the trace of a meridian is constant. We also prove that the adjoint Reidemeister torsion of $K$ satisfies the vanishing identity if each $K_i$ does so.

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arXiv2021

A vanishing identity on adjoint Reidemeister torsions of twist knots

For a compact oriented 3-manifold with torus boundary the adjoint Reidemeister torsion is defined as a function on the $\mathrm{SL}_2(\mathbb{C})$-character variety depending on a choice of a boundary curve. Under reasonable assumptions, it is conjectured that the adjoint torsion satisfies a certain type of vanishing identities. In this paper, we prove that the conjecture holds for all hyperbolic twist knot exteriors by using Jacobi's residue theorem.

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arXiv2019

Adjoint Reidemeister torsions from wrapped M5-branes

We introduce a vanishing property of adjoint Reidemeister torsions of a cusped hyperbolic 3-manifold derived from the physics of wrapped M5-branes on the manifold. To support our physical observation, we present a rigorous proof for the figure-eight knot complement with respect to all slopes. We also present numerical verification for several knots.

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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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arXiv2019

The volume and Chern-Simons invariant of a Dehn-filled manifold

For a compact 3-manifold $N$ with non-empty boundary, Zickert gave a combinatorial formula for computing the volume and Chern-Simons invariant of a boundary parabolic representation $π_1(N)\rightarrow \mathrm{PSL}(2,\mathbb{C})$. In this paper, we introduce a notion of deformed Ptolemy varieties and extend the formula of Zickert to a representation that is not necessarily boundary parabolic. This allows us to compute the volume and Chern-Simons invariant of a $\mathrm{PSL}(2,\mathbb{C})$-representation of a closed 3-manifold.

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