arXiv subjects

Ryan Blair

Publications and source records attributed to Ryan Blair.

At least 19 recordsLinked to original sources

arXiv2026

Sharp Nordhaus-Gaddum bounds for throttling

Throttling is a graph optimization problem, where the throttling number of a graph is the minimum sum or minimum product of the number of vertices in an initial set and the time required to complete a certain graph operation. A Nordhaus-Gaddum bound refers to an upper or lower bound of the sum or product of a graph parameter together with that of its complement. In this paper, we study the Nordhaus-Gaddum sum and product bounds of the various throttling numbers (sum throttling and product throttling with or without initial cost). Graph operations considered are standard zero forcing, positive semidefinite forcing, power domination, and Cops and Robbers.

math.CO↗

arXiv2026

Entanglement complexity of spanning pairs of lattice polygons

We study the entanglement complexity of a system consisting of two simple-closed curves (self-avoiding polygons) that span a lattice tube, referred to as a 2SAP. 2SAPs are of interest as the first known model of confined ring polymers where the linking probability goes to 1 exponentially with the size of the system. Atapour et al proved this in 2010 by showing that all but exponentially few sufficiently large 2SAPs contain a pattern that guarantees the 2SAP is non-split, provided that the requisite pattern fits in the tube. This result was recently extended to all tubes sizes that admit non-trivial links. Here we develop and apply knot theory results to answer more general questions about the entanglement complexity of 2SAPs. We first extend the 1992 concept of a good measure of knot complexity to a good measure, $F$, of spanning-link complexity for $k$-component links. Using tangle products, we show, for example, that the more complex the prime knot decomposition of any component of a given link type, the greater its $F$-measure. We then prove that all but exponentially few size $m$ 2SAPs have $F$ complexity that grows at least linearly in $m$ as $m\to \infty$. We establish that good measures of knot complexity yield good measures of spanning-link complexity. We also establish conditions whereby more general link invariants can yield good measures. In particular, we establish that measures based on several classical invariants are good measures by our definition, eg bridge number or the splitting number. Finally, we consider how the tube dimensions affect which links are embeddable as 2SAPs as well as geometric restrictions on the entanglement complexity of the embeddings. For example, we establish that there are two-component links that occur as 2SAPs in a given tube size only when one of the components is forced into a non-minimal bridge number conformation.

math.GT↗

arXiv2025

Rep-Tiles

An $n$-dimensional rep-tile is a compact, connected submanifold of $\mathbb{R}^n$ with non-empty interior which can be decomposed into pairwise isometric rescaled copies of itself whose interiors are disjoint. We show that every smooth compact $n$-dimensional submanifold of $\mathbb{R}^n$ with connected boundary is topologically isotopic to a polycube that tiles the $n$-cube, and hence is topologically isotopic to a rep-tile. It follows that there is a rep-tile in the homotopy type of any finite CW complex. In addition to classifying rep-tiles in all dimensions up to isotopy, we also give new explicit constructions of rep-tiles, namely examples in the homotopy type of any finite bouquet of spheres.

math.GT↗

arXiv2025

Efficient Detection of Borromean Linking in Ellipses

We describe a method by which the number of intersections one ellipse makes inside the plane of another can be determined. The method is based on applying a transformation that reverts one ellipse to the unit circle, and examining the intersection points of the transformed second ellipse with the unit circle. This may be used to efficiently determine Hopf linking between two ellipses, or Borromean linking between three ellipses.

math.GT↗

arXiv2025

The Plain Sphere Number of a Link

Let $L$ be a link in $S^3$. We consider the class of meridional presentations for $π_1(S^3\backslash L)$ in which the relations are witnessed by embedded two-spheres which can be represented simultaneously in a fixed diagram of $L$, analogously to decomposition spheres studied by Cromwell, Menasco and others. Wirtinger relations are witnessed by such spheres and the Wirtinger presentation is a special case of the ones we study. We prove that the smallest number of generators of $π_1(S^3\backslash L)$ over all such presentations equals the bridge number of $L$.

math.GT↗

arXiv2025

Sharp bounds for product and sum throttling numbers

Throttling in graphs optimizes a sum or product of resources used, such as the number of vertices in an initial set, and time required, such as the propagation time, to complete a given task. We introduce a new technique to establish sharp upper bounds in terms of graph order for sum throttling and initial cost product throttling for power domination. Furthermore, we establish sharp bounds on possible changes of the product throttling number, both with and without initial cost, caused by certain graph operations for standard zero forcing, positive semidefinite forcing, and power domination.

math.CO↗

arXiv2024

Adding a suitable unknot to any link equates bridge number and meridional rank

Given any link $L\subseteq S^3$, we show that it is possible to embed an unknot $U$ in its complement so that the link $L\cup U$ satisfies the Meridional Rank Conjecture (MRC). The bridge numbers in our construction fit into the equality $β(L\cup U)=2β(L)-1=\text{rank}(π_1(S^3\backslash (L\cup U)))$. In addition, we prove the MRC for new infinite families of links and distinguish them from previously settled cases through an application of bridge distance.

math.GT↗

arXiv2022

Coxeter quotients of knot groups through 16 crossings

We find explicit maximal rank Coxeter quotients for the knot groups of 595,515 out of the 1,701,936 knots through 16 crossings. We thus calculate the bridge numbers and verify Cappell and Shaneson's Meridional Rank Conjecture for these knots. In addition, we provide a computational tool for establishing the conjecture for knots beyond 16 crossings whose meridional ranks can be detected via finite Coxeter quotients.

math.GT↗

arXiv2022

Percolation and Dissolution of Borromean Networks

Inspired by experiments on topologically linked DNA networks, we consider the connectivity of Borromean networks, in which no two rings share a pairwise-link, but groups of three rings form inseparable triplets. Specifically, we focus on square lattices at which each node is embedded a loop which forms a Borromean link with pairs of its nearest neighbors. By mapping the Borromean link network onto a lattice representation, we investigate the percolation threshold of these networks, (the fraction of occupied nodes required for a giant component), as well as the dissolution properties: the spectrum of topological links that would be released if the network were dissolved to varying degrees. We find that the percolation threshold of the Borromean square lattice occurs when approximately 60.75\% of nodes are occupied, slightly higher than the 59.27\% typical of a square lattice. Compared to the dissolution of Hopf-linked networks, a dissolved Borromean network will yield more isolated loops, and fewer isolated triplets per single loop. Our simulation results may be used to predict experiments from Borromean structures produced by synthetic chemistry.

cond-mat.stat-mech↗

arXiv2021

Three-dimensional Rep-tiles

A 3D rep-tile is a compact 3-manifold $X$ in $\mathbb{R}^3$ that can be decomposed into finitely many pieces, each of which are similar to $X$, and all of which are congruent to each other. In this paper we classify all 3D rep-tiles up to homeomorphism. In particular, we show that a 3-manifold is homeomorphic to a 3D rep-tile if and only if it is the exterior of a connected graph in $S^3$.

math.GT↗

arXiv2021

Kirby-Thompson distance for trisections of knotted surfaces

We adapt work of Kirby-Thompson and Zupan to define an integer invariant $\mathcal{L}(\mathcal{T})$ of a bridge trisection $\mathcal{T}$ of a smooth surface $\mathcal{K}$ in $S^4$ or $B^4$. We show that when $\mathcal{L}(\mathcal{T})=0$, then the surface $\mathcal{K}$ is unknotted. We also show show that for a trisection $\mathcal{T}$ of an irreducible surface, bridge number produces a lower bound for $\mathcal{L}(\mathcal{T})$. Consequently, $\mathcal{L}$ can be arbitrarily large.

math.GT↗

arXiv2021

Self-replicating 3-manifolds

In this paper we explore the topological properties of self-replicating, 3-dimensional manifolds, which are modeled by idempotents in the (2+1)-cobordism category. We give a classification theorem for all such idempotents. Additionally, we characterize biologically interesting ways in which self-replicating 3-manifolds can embed in $\mathbb{R}^3$.

math.GT↗

arXiv2020

Coxeter groups and meridional rank of links

We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on meridional rank are obtained via Coxeter quotients of the groups of link complements. Matching upper bounds on bridge number are found using the Wirtinger numbers of link diagrams, a combinatorial tool developed by the authors.

math.GT↗

arXiv2020

A note on three-fold branched covers of $S^4$

We show that any 4-manifold admitting a $(g;k_1,k_2,0)$-trisection is an irregular 3-fold cover of the 4-sphere whose branching set is a surface in $S^4$, smoothly embedded except for one singular point which is the cone on a link. A 4-manifold admits such a trisection if and only if it has a handle decomposition with no 1-handles; it is conjectured that all simply-connected 4-manifolds have this property.

math.GT↗

arXiv2020

The bridge number of arborescent links with many twigs

We prove the meridional rank conjecture for arborescent links associated to plane trees with the following property: all branching points carry a straight branch to at least three leaves. The proof involves an upper bound on the bridge number in terms of the maximal number of link components of the underlying tree, valid for all arborescent links.

math.GT↗

arXiv2019

Distortion and the bridge distance of knots

We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of knots for which the minimum of the bridge distance and the bridge number is unbounded and Pardon's lower bound is constant.

math.GT↗

arXiv2019

Knots with Exactly 10 Sticks

We prove that the knots $13n_{592}$ and $15n_{41,127}$ both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known.

math.GT↗

arXiv2019

The US Program in Ground-Based Gravitational Wave Science: Contribution from the LIGO Laboratory

Recent gravitational-wave observations from the LIGO and Virgo observatories have brought a sense of great excitement to scientists and citizens the world over. Since September 2015,10 binary black hole coalescences and one binary neutron star coalescence have been observed. They have provided remarkable, revolutionary insight into the "gravitational Universe" and have greatly extended the field of multi-messenger astronomy. At present, Advanced LIGO can see binary black hole coalescences out to redshift 0.6 and binary neutron star coalescences to redshift 0.05. This probes only a very small fraction of the volume of the observable Universe. However, current technologies can be extended to construct "$3^\mathrm{rd}$ Generation" (3G) gravitational-wave observatories that would extend our reach to the very edge of the observable Universe. The event rates over such a large volume would be in the hundreds of thousands per year (i.e.tens per hour). Such 3G detectors would have a 10-fold improvement in strain sensitivity over the current generation of instruments, yielding signal-to-noise ratios of 1000 for events like those already seen. Several concepts are being studied for which engineering studies and reliable cost estimates will be developed in the next 5 years.

astro-ph.IM↗