arXiv subjects

Alexander R. Klotz

Publications and source records attributed to Alexander R. Klotz.

At least 19 recordsLinked to original sources

arXiv2026

Pearl necklace knots with fewer vertices than an equivalent FCC lattice knot

The pearl necklace number, $N_P$, of a knot is the smallest number of unit spheres required to construct a knot if each sphere is tangent to two neighbors and no sphere overlaps with another. It has been speculated that the pearl necklace number is equal to the minimum number of lattice sites required to embed a knot on a face centered cubic (FCC) lattice $N_L$, implying that a trefoil knot cannot be constructed from fewer than 15 spheres. A large language model (LLM) was prompted to find configurations of knot for which $N_P<N_L$, and this manuscript describes its findings, attempts at validating them, and their implications. No 14-vertex example was found for the trefoil knot, but examples with $N_P=N_L-1$ were found for all knots from 5 to 7 crossings as well as $8_{19}$, and $10_{124}$. One example, $8_1$, could be constructed with $N_P=N_L-2$.

math.GT↗

arXiv2026

A solid-state theory for dense cylindrical packings of balls

We develop an analytical theory for the dense packing of hard spheres in cylinders. Physically, our theory consists of a finite cylindrical masking of a close-packed three-dimensional solid and covers the entire range of cylinder aspect ratios, thus going beyond efforts that are focused on very tall cylinders in a narrow range of widths. We explicitly derive an exact equation for resulting packing fractions, valid for any regular lattice, and it provides a basis to understand the oscillations and scaling of volume fractions that have appeared in previous works. Our analytical relation serves as a rigorous lower bound and to tighten it we derive, and implement, an efficient mathematical procedure to optimize the orientation of the cylinder. Furthermore, we suggest simple techniques to improve on the predicted packings. Overall, we provide a general theoretical foundation for the packing of balls in cylinders, valid for all container sizes.

cond-mat.soft↗

arXiv2026

Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links

Ropelength, L, is a parameter characterizing the minimum contour length of a knot or link. There exist upper and lower bounds on ropelength with respect to crossing number, C, including a universal lower bound constraining $L\geqα_0 C^{3/4}$ for some constant $α_0$. There is currently an order-of-magnitude range for the value of $α_0$ between 1.105 and 10.76. In this work, we show that T(Q,Q) torus links can be constructed such that the upper bound is within a factor of 1.77 of the lower bound. We derive a stronger lower bound based on the convex hull around close-packed disks of approximately $α_{T_{QQ}}>\sqrt{8π\sqrt{3}}+(2π+\sqrt{2π+7\sqrt{3}-12}\ )Q^{-1/2}\approx6.60+7.61Q^{-1/2}$, significantly higher than the best universal lower bound of 1.105. We show that a link can be constructed without any free parameters or geometric optimization that, when $Q$ is large, has a coefficient $α_{T_{QQ}}<1.005\cdot 4π(5\sqrt{5}-8)/3\approx13.39$, and can be improved to to 11.68 by solving a helical no-overlap constraint equation that requires a conjectural approximation. For $Q$ up to 20 we construct links from smooth planar curves or toroidal helices minimized with respect to a small number of geometric parameters, that are between 6 and 60% greater in ropelength than the lower bound. Many such links can be annealed to within 10% of the lower bound using gradient descent. This represents significant progress towards developing sharp bounds on the ropelengths of specific classes of knots and links.

math.GT↗

arXiv2026

Exactly-solvable self-trapping lattice walks. II. Lattices of arbitrary height

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and has a trapped endpoint. We show that the generating function enumerating GSAWs on a half-infinite strip of finite height is rational, and we give a procedure to construct a combinatorial finite state machine that allows one to compute this generating function. We then modify this procedure to compute generating functions for GSAWs under two probabilistic models. We perform Monte Carlo simulations to estimate the expected length and displacement for GSAWs on the quarter plane, half plane, full plane, and half-infinite strips of bounded height for which we cannot compute the generating function. Finally, we prove that the generating functions for Greek key tours (GSAWs on a finite grid that visit every vertex) on a half-infinite strip of fixed height are also rational, allowing us to resolve several conjectures.

math.CO↗

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

Topologically Directed Simulations Reveal the Impact of Geometric Constraints on Knotted Proteins

Simulations of knotting and unknotting in polymers or other filaments rely on random processes to facilitate topological changes. Here we introduce a method of \textit{topological steering} to determine the optimal pathway by which a filament may knot or unknot while subject to a given set of physics. The method involves measuring the knotoid spectrum of a space curve projected onto many surfaces and computing the mean unravelling number of those projections. Several perturbations of a curve can be generated stochastically, e.g. using the Langevin equation or crankshaft moves, and a gradient can be followed that maximises or minimises the topological complexity. We apply this method to a polymer model based on a growing self-avoiding tangent-sphere chain, which can be made to model proteins by imposing a constraint that the bending and twisting angles between successive spheres must maintain the distribution found in naturally occurring protein structures. We show that without these protein-like geometric constraints, topologically optimised polymers typically form alternating torus knots and composites thereof, similar to the stochastic knots predicted for long DNA. However, when the geometric constraints are imposed on the system, the frequency of twist knots increases, similar to the observed abundance of twist knots in protein structures.

math.GT↗

arXiv2025

Ropelength-minimizing concentric helices and non-alternating torus knots

An alternating torus knot or link may be constructed from a repeating double helix after connecting its two ends. A structure with additional helices may be closed to form a non-alternating torus knot or link. Previous work has optimized the dimensions and pitch of double helices to derive upper bounds on the ropelength of alternating torus knots, but non-alternating knots have not been studied extensively and are known to be tighter. Here, we examine concentric helices as units of non-alternating torus knots and discuss considerations for minimizing their contour length. By optimizing both the geometry and combinatorics of the helices, we find efficient configurations for systems with between 3 and 39 helices. Using insights from those cases, we develop an efficient construction for larger systems and show that concentric helices distributed between many shells have an optimized ropelength of approximately 7.83Q^(3/2) where Q is the total number of helices or the minor index of the torus knot, and the prefactor is exact and a 75 percent reduction from previous work. Links formed by extending these helices and bending them into a T(3Q,Q) torus link have a ropelength that is approximately 12 times the three-quarter power of the crossing number. These results reduce the ratio between the upper and lower bounds of the ropelength of non-alternating torus knots from 29 to between 1.4 and 3.8.

math.GT↗

arXiv2025

The space writhes and signatures of polymer knots

The space writhe of a knot is a property of its three-dimensional embedding that contains information about its underlying topology, but the correspondence between space writhe and other topological invariants is not fully understood. We perform Langevin dynamics simulations of knotted semiflexible polymers and measure their ensemble average space writhe. We show that for all knots up to 10 crossings, alternating and non-alternating, the average space writhe is almost equal to that of the tightest known configuration of the same knot, with minor differences. Using this equivalence, we show that for more complex knots with up to 38 crossings, the average space writhe is strongly correlated with the signature of the knot. This establishes that the connection between signature and space writhe holds at larger crossing numbers.

cond-mat.soft↗

arXiv2024

The Gravity Tunnel Superhighway

This manuscript discusses gravity tunnels formed by connecting two vertical shafts by a constant-radius tunnel within the Earth, which featured in a dream I had in September 2024. The total travel time through such a tunnel can be minimized with respect to the radius at which the shafts are connected. I derive this minimal radius and minimum time given two assumptions for Earth's interior, that of constant gravitational acceleration and that of uniform density. Both models have solutions in terms of basic functions, and are typically 10% slower than the brachistochrone curve between the same points. I also find the optimal depth of a "superhighway," which minimizes the average time to fall between any two points on a great circle. Finally, I discuss the role of problems like these in physics education.

physics.pop-ph↗

arXiv2024

Dynamics of polymers in coarse-grained nematic solvents

Polymers are a primary building block in many biomaterials, often interacting with anisotropic backgrounds. While previous studies have considered polymer dynamics within nematic solvents, rarely are the the effects of anisotropic viscosity and polymer elongation differentiated. Here, we study polymers embedded in nematic liquid crystals with isotropic viscosity via numerical simulations, to explicitly investigate the effect of nematicity on macromolecular conformation and how conformation alone can produce anisotropic dynamics. We employ a hybrid technique that captures nematic orientation, thermal fluctuations and hydrodynamic interactions. The coupling of the polymer backbone to the nematic field elongates the polymer, producing anisotropic diffusion even in nematic solvents with isotropic viscosity. For intermediate coupling, the competition between background anisotropy and macrmolecular entropy leads to hairpins - sudden kinks along the backbone of the polymer. Experiments of DNA embedded in a solution of rod-like fd viruses qualitatively support the role of hairpins in establishing characteristic conformational features that govern polymer dynamics. Hairpin diffusion along the backbone exponentially slows as coupling increases. Better understanding two-way coupling between polymers and their surroundings could allow the creation of more biomimetic composite materials.

cond-mat.soft↗

arXiv2024

Chirality Effects in Molecular Chainmail

Motivated by the observation of positive Gaussian curvature in kinetoplast DNA networks, we consider the effect of linking chirality in square lattice molecular chainmail networks using Langevin dynamics simulations and constrained gradient optimization. Linking chirality here refers to ordering of over-under versus under-over linkages between a loop and its neighbors. We consider fully alternating linking, maximally non-alternating, and partially non-alternating linking chiralities. We find that in simulations of polymer chainmail networks, the linking chirality dictates the sign of the Gaussian curvature of the final state of the chainmail membranes. Alternating networks have positive Gaussian curvature, similar to what is observed in kinetoplast DNA networks. Maximally non-alternating networks form isotropic membranes with negative Gaussian curvature. Partially non-alternating networks form flat diamond-shaped sheets which undergo a thermal folding transition when sufficiently large, similar to the crumpling transition in tethered membranes. We further investigate this topology-curvature relationship on geometric grounds by considering the tightest possible configurations and the constraints that must be satisfied to achieve them.

cond-mat.soft↗

arXiv2024

Borromean Hypergraph Formation in Dense Random Rectangles

We develop a minimal system to study the stochastic formation of Borromean links within topologically entangled networks without requiring the use of knot invariants. Borromean linkages may form in entangled solutions of open polymer chains or in Olympic gel systems such as kinetoplast DNA, but it is challenging to investigate this due to the difficulty of computing three-body link invariants. Here, we investigate randomly oriented rectangles densely packed within a volume, and evaluate them for Hopf linking and Borromean link formation. We show that dense packings of rectangles can form Borromean triplets and larger clusters, and that in high enough density the combination of Hopf and Borromean linking can create a percolating hypergraph through the network. We present data for the percolation threshold of Borromean hypergraphs, and discuss implications for the existence of Borromean connectivity within kinetoplast DNA.

cond-mat.soft↗

arXiv2024

Revisiting the Second Vassiliev (In)variant for Polymer Knots

Knots in open strands such as ropes, fibers, and polymers, cannot typically be described in the language of knot theory, which characterizes only closed curves in space. Simulations of open knotted polymer chains, often parameterized to DNA, typically perform a closure operation and calculate the Alexander polynomial to assign a knot topology. This is limited in scenarios where the topology is less well-defined, for example when the chain is in the process of untying or is strongly confined. Here, we use a discretized version of the Second Vassiliev Invariant for open chains to analyze Langevin Dynamics simulations of untying and strongly confined polymer chains. We demonstrate that the Vassiliev parameter can accurately and efficiently characterize the knotted state of polymers, providing additional information not captured by a single-closure Alexander calculation. We discuss its relative strengths and weaknesses compared to standard techniques, and argue that it is a useful and powerful tool for analyzing polymer knot simulations.

cond-mat.soft↗

arXiv2023

Linear Polycatenanes from Kinetoplast Edge Loops

We use graph theory simulations and single molecule experiments to investigate percolation properties of kinetoplasts, the topologically linked mitochondrial DNA from trypanosome parasites. The edges of some kinetoplast networks contain a fiber of redundantly catenated DNA loops, but previous investigations of kinetoplast topology did not take this into account. Our graph simulations track the size of connected components in lattices as nodes are removed, analogous to the removal of minicircles from kinetoplasts. We find that when the edge loop is taken into account, the largest component after the network de-percolates is a remnant of the edge loop, before it undergoes a second percolation transition and breaks apart. This implies that stochastically removing minicircles from kinetoplast DNA would isolate large polycatenanes, which is observed in experiments that use photonicking to stochastically destroy kinetoplasts from Crithidia fasciculata. Our results imply kinetoplasts may be used as a source of linear polycatenanes for future experiments.

cond-mat.soft↗

arXiv2023

Ropelength and writhe quantization of 12-crossing knots

The ropelength of a knot is the minimum length required to tie it. Computational upper bounds have previously been computed for every prime knot with up to 11 crossings. Here, we present ropelength measurements for the 2176 knots with 12 crossings, of which 1288 are alternating and 888 are non-alternating. We report on the distribution of ropelengths within and between crossing numbers, as well as the space writhe of the tight knot configurations. It was previously established that tight alternating knots have a ``quantized'' space writhe close to a multiple of 4/7. Our data supports this for 12-crossing alternating knots and we find that non-alternating knots also show evidence of writhe quantization, falling near integer or half-integer multiples of 4/3, depending on the parity of the crossing number. Finally, we examine correlations between geometric properties and topological invariants of tight knots, finding that the ropelength is positively correlated with hyperbolic volume and its correlates, and that the space writhe is positively correlated with the Rasmussen s invariant.

math.GT↗

arXiv2022

Exactly-Solvable Self-Trapping Lattice Walks. Part I: Trapping in Ladder Graphs

A growing self-avoiding walk (GSAW) is a stochastic process that starts from the origin on a lattice and grows by occupying an unoccupied adjacent lattice site at random. A sufficiently long GSAW will reach a state in which all adjacent sites are already occupied by the walk and become trapped, terminating the process. It is known empirically from simulations that on a square lattice, this occurs after a mean of 71 steps. In Part I of a two-part series of manuscripts, we consider simplified lattice geometries only two sites high ("ladders") and derive generating functions for the probability distribution of GSAW trapping. We prove that a self-trapping walk on a square ladder will become trapped after a mean of 17 steps, while on a triangular ladder trapping will occur after a mean of 941/48 (~19.6 steps). We discuss additional implications of our results for understanding trapping in the "infinite" GSAW.

math.CO↗

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

The tightest knot is not necessarily the smallest

In this note, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its convex hull. We measure the convex hull volume of knots during the length annealing process, identifying local minima in the hull volume that arise due to buckling and symmetry breaking. We use T(p,2) torus knots as an illustrative example of a family of knots whose locally minimal-length embeddings are not necessarily ordered by volume. We identify several knots whose central curve has a convex hull volume that is not minimized in the ideal configuration, and find that $8_{19}$ has a non-ideal global minimum in its convex hull volume even when the thickness of its tube is taken into account.

math.GT↗