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Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked patterns, which we refine into precise conjectures. We then prove several results along those lines, for example that generically at least 69% of the roots are on the unit circle, which appears to be sharp. We also show there is a large root-free region near the origin. We further study the equidistribution properties of such roots by introducing a Lyapunov exponent of the Burau representation of random positive braids, and a corresponding bifurcation measure. In the spirit of Deroin and Dujardin, we conjecture that the bifurcation measure gives the limiting measure for such roots, and prove this on a region with positive limiting mass. We use tools including work of Gambaudo and Ghys on the signature function of links, for which we prove a central limit theorem. 

Systematic enumeration and identification of unique 3D spatial topologies (STs) of complex engineering systems (such as automotive cooling systems, electric power trains, satellites, and aero-engines) are essential to navigation of these expansive design spaces with the goal of identifying new spatial configurations that can satisfy challenging system requirements. However, efficient navigation through discrete 3D ST options is a very challenging problem due to its combinatorial nature and can quickly exceed human cognitive abilities at even moderate complexity levels. This article presents a new, efficient, and scalable design framework that leverages mathematical spatial graph theory to represent, enumerate, and identify distinctive 3D topological classes for a generic 3D engineering system, given its system architecture (SA)—its components and their interconnections. First, spatial graph diagrams (SGDs) are generated for a given SA from zero to a specified maximum number of interconnect crossings. Then, corresponding Yamada polynomials for all the planar SGDs are generated. SGDs are categorized into topological classes, each of which shares a unique Yamada polynomial. Finally, within each topological class, 3D geometric models are generated using the SGDs having different numbers of interconnect crossings. Selected case studies are presented to illustrate the different features of our proposed framework, including an industrial engineering design application: ST enumeration of a 3D automotive fuel cell cooling system (AFCS). Design guidelines are also provided for practicing engineers to aid the application of this framework to different types of real-world problems such as configuration design and spatial packaging optimization. 

Code and data to accompany the paper of the same name. 

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture. 

Systematic enumeration and identification of unique 3D spatial topologies of complex engineering systems such as automotive cooling layouts, hybrid-electric power trains, and aero-engines are essential to search their exhaustive design spaces to identify spatial topologies that can satisfy challenging system requirements. However, efficient navigation through discrete 3D spatial topology options is a very challenging problem due to its combinatorial nature and can quickly exceed human cognitive abilities at even moderate complexity levels. Here we present a new, efficient, and generic design framework that utilizes mathematical spatial graph theory to represent, enumerate, and identify distinctive 3D topological classes for an abstract engineering system, given its system architecture (SA) — its components and interconnections. Spatial graph diagrams (SGDs) are generated for a given SA from zero to a specified maximum crossing number. Corresponding Yamada polynomials for all the planar SGDs are then generated. SGDs are categorized into topological classes, each of which shares a unique Yamada polynomial. Finally, for each topological class, one 3D geometric model is generated for an SGD with the fewest interconnect crossings. Several case studies are shown to illustrate the different features of our proposed framework. Design guidelines are also provided for practicing engineers to aid the utilization of this framework for application to different types of real-world problems. 

Code and data to accompany the paper of the same name by N. M. Dunfield, S. Garoufalidis, and J. H. Rubinstein. 

This paper describes the complete list of all 205,822 exceptional Dehn fillings on the 1-cusped hyperbolic 3-manifolds that have ideal triangulations with at most 9 ideal tetrahedra. The data is consistent with the standard conjectures about Dehn filling and suggests some new ones. 

Code and data to accompany the paper of the same name.