arXiv subjects

Connie On Yu Hui

Publications and source records attributed to Connie On Yu Hui.

9 recordsLinked to original sources

arXiv2026

The Lorenz braid index and hyperbolic volume

A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.

math.GT↗

arXiv2026

Generalised T-links Represent All Links in the 3-Sphere

We introduce generalised T-links, a restricted extension of the widely studied families of T-links and twisted torus links, and prove that this family is universal: every link in the $3$-sphere can be represented as a generalised T-link. This gives a compact braid-theoretic form of Ghrist's universality theorem for certain Lorenz-like templates. More precisely, we replace the template-theoretic representatives coming from universal Lorenz-like templates with explicit braid closures depending on finitely many integer parameters. Although closely related braid families have appeared previously in the literature, the universality of this restricted class is the main point of the present paper. Thus, all links in the $3$-sphere are placed inside a natural structured class closely related to T-links, Lorenz links, and twisted torus links.

math.GT↗

arXiv2025

A robot that unknots knots

Consider a robot that remembers only the starting position and walks along a knot once on a knot diagram, switching every undercrossing it meets until it returns to the starting position. We observe that the robot produces an ascending diagram, and we provide a new combinatorial proof that every ascending or descending knot diagram can be transformed into the zero-crossing unknot diagram. Using the machinery developed from the combinatorial proof, we show that the minimal number of Reidemeister moves required for such a transformation is bounded above by (7C+1)C if the diagram has C crossings. Moreover, we provide a new alternative proof that there exist sequences of Reidemeister moves that do not increase the number of crossings and transform ascending or descending knot diagrams into zero-crossing unknot diagrams.

math.GT↗

arXiv2025

Volume bounds for hyperbolic rod complements in the 3-torus

The study of rod complements is motivated by rod packing structures in crystallography. We view them as complements of links comprised of Euclidean geodesics in the 3-torus. Recent work of the second author classifies when such rod complements admit hyperbolic structures, but their geometric properties are yet to be well understood. In this paper, we provide upper and lower bounds for the volumes of all hyperbolic rod complements in terms of rod parameters, and show that these bounds may be loose in general. We introduce better and asymptotically sharp volume bounds for a family of rod complements. The bounds depend only on the lengths of the continued fractions formed from the rod parameters.

math.GT↗

arXiv2025

Modular links: Bunch algorithm and upper volume bounds

In the 1970s, Williams developed an algorithm that has been used to construct and study modular links in the Lorenz template. We introduce an improved algorithm, which we call the bunch algorithm, to provide more insights into the geometry of modular links and Lorenz links. Using the machinery developed for the bunch algorithm, we provide the first upper volume bound that is independent of word exponents and quadratic in the braid index of the Lorenz link component for all modular link complements. We find families of modular knot complements with upper volume bounds that are linear in the braid index. A classification of modular link complements based on the relative magnitudes of word exponents is also presented.

math.GT↗

arXiv2024

A lower bound for the volumes of modular link complements

Every finite collection of oriented closed geodesics in the modular surface has a canonically associated link in its unit tangent bundle coming from the periodic orbits of the geodesic flow. We study the volume of the associated link complement with respect to its unique complete hyperbolic metric. We provide the first lower volume bound that is linear in terms of the number of distinct exponents in the code words corresponding to the collection of closed geodesics.

math.GT↗

arXiv2024

Crushing Surfaces of Positive Genus

The operation of crushing a normal surface has proven to be a powerful tool in computational $3$-manifold topology, with applications both to triangulation complexity and to algorithms. The main difficulty with crushing is that it can drastically change the topology of a triangulation, so applications to date have been limited to relatively simple surfaces: $2$-spheres, discs, annuli, and closed boundary-parallel surfaces. We give the first detailed analysis of the topological effects of crushing closed essential surfaces of positive genus. To showcase the utility of this new analysis, we use it to prove some results about how triangulation complexity interacts with JSJ decompositions and satellite knots; although similar applications can also be obtained using techniques of Matveev, our approach has the advantage that it avoids the machinery of almost simple spines and handle decompositions.

math.GT↗

arXiv2024

A geometric classification of rod complements in the 3-torus

Rod packings are used in crystallography to describe crystal structures with linear or zigzag chains of particles, and each rod packing can be topologically viewed as a collection of disjoint geodesics in the 3-torus. Hui and Purcell developed a method to study the complements of rods in the 3-torus with the use of 3-dimensional geometry and tools from the 3-sphere, and they partially classified the geometry of some families of rod complements in the 3-torus. In this paper, we provide a complete classification of the geometry of all rod complements in the 3-torus using topological arguments.

math.GT↗

arXiv2023

On the geometry of rod packings in the 3-torus

Rod packings in the 3-torus encode information of some crystal structures in crystallography. They can be viewed as links in the 3-torus, and tools from 3-manifold geometry and topology can be used to study their complements. In this paper, we initiate the use of geometrisation to study such packings. We analyse the geometric structures of the complements of simple rod packings, and find families that are hyperbolic and Seifert fibred.

math.GT↗