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Yunhi Cho

Publications and source records attributed to Yunhi Cho.

10 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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arXiv2024

Uniqueness of non-Euclidean Mass Center System and Generalized Pappus' Centroid Theorems in Three Geometries

G.A. Galperin introduced the axiomatic mass center system for finite point sets in spherical and hyperbolic spaces, proving the uniqueness of the mass center system. In this paper, we revisit this system and provide a significantly simpler proof of its uniqueness. Furthermore, we extend the axiomatic mass center system to manifolds. As an application of our system, we derive a highly generalized version of Pappus' centroid theorem for volumes in three geometries - Euclidean, spherical, and hyperbolic - across all dimensions, offering unified and notably simple proofs for all three geometries.

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arXiv2023

Rigidity of nonconvex polyhedra with respect to edge lengths and dihedral angles

We prove that every three-dimensional polyhedron is uniquely determined by its dihedral angles and edge lengths, even if nonconvex or self-intersecting, under two plausible sufficient conditions: (i) the polyhedron has only convex faces and (ii) it does not have partially-flat vertices, and under an additional technical requirement that (iii) any triple of vertices is not collinear. The proof is consistently valid for Euclidean, hyperbolic and spherical geometry, which takes a completely different approach from the argument of the Cauchy rigidity theorem. Various counterexamples are provided that arise when these conditions are violated, and self-contained proofs are presented whenever possible. As a corollary, the rigidity of several families of polyhedra is also established. Finally, we propose two conjectures: the first suggests that Condition (iii) can be removed, and the second concerns the rigidity of spherical nonconvex polygons.

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arXiv2022

Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities

There is an elegant expression for the volume of hypercube $[0,1]^n$ clipped by a single hyperplane. In the article the formula is generalized to the case of more than one hyperplane. An important foundation for the result is Lawrence's formula and a way to weaken two restrictions of simplicity and non-parallelness in his formula is also considered. Several concrete volume formulas of clipped hypercubes are derived explicitly and the corresponding combinatorial identities are obtained as an application.

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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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arXiv2017

Partially abelian representations of knot groups

A knot complement admits a pseudo-hyperbolic structure by solving Thurston's gluing equations for an octahedral decomposition. It is known that a solution to these equations can be described in terms of region variables, also called $w$-variables. In this paper, we consider the case when pinched octahedra appear as a boundary parabolic solution in the decomposition. A $w$-solution with pinched octahedra induces a solution for a new knot obtained by changing the crossing or inserting a tangle at the pinched place. We discuss this phenomenon with corresponding holonomy representations and give some examples including ones obtained from connected sum.

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arXiv2010

The analytic continuation of hyperbolic space

We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended space gives rise to a complex valued geometry consistent with both the hyperbolic and de Sitter space. Such a construction shed a light and inspires a new insight for the study of the hyperbolic geometry and Lorentzian geometry. We discuss the advantages of this new geometric model as well as some of its applications.

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arXiv2007

Trigonometry in extended hyperbolic space and extended de Sitter space

We study the hyperbolic cosine and sine laws in the extended hyperbolic space which contains hyperbolic space as a subset and is an analytic continuation of the hyperbolic space. And we also study the spherical cosine and sine laws in the extended de Sitter space which contains de Sitter Space $S^n_1$ as a subset and is also an analytic continuation of de Sitter space. In fact, the extended hyperbolic space and extended de Sitter space are the same space only differ by -1 multiple in the metric. Hence these two extended spaces clearly show and apparently explain that why many corresponding formulas in hyperbolic and spherical space are very similar each other. From these extended trigonometry laws, we can give a coherent and geometrically simple explanation for the various relations between the lengths and angles of hyperbolic polygons and relations on de Sitter polygons which lie on $S^2_1$.

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arXiv2006

Volume of $C^{1,α}$-boundary domain in extended hyperbolic space

We consider the projectivization of Minkowski space with the analytic continuation of the hyperbolic metric and call this an extended hyperbolic space. We can measure the volume of a domain lying across the boundary of the hyperbolic space using an analytic continuation argument. In this paper we show this method can be further generalized to find the volume of a domain with smooth boundary with suitable regularity in dimension 2 and 3. We also discuss that this volume is invariant under the group of hyperbolic isometries and that this regularity condition is sharp.

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