Almost positive links are strongly quasipositive

We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow’s in the (strong) positive. As a second main result, we give a simple and complete characterization of link diagrams with quasipositive canonical surface (the surface produced by Seifert’s algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.

Dynamic graph exploration by interactively linked node-link diagrams and matrix visualizations

The visualization of dynamic graphs is a challenging task owing to the various properties of the underlying relational data and the additional time-varying property. For sparse and small graphs, the most efficient approach to such visualization is node-link diagrams, whereas for dense graphs with attached data, adjacency matrices might be the better choice. Because graphs can contain both properties, being globally sparse and locally dense, a combination of several visual metaphors as well as static and dynamic visualizations is beneficial. In this paper, a visually and algorithmically scalable approach that provides views and perspectives on graphs as interactively linked node-link and adjacency matrix visualizations is described. As the novelty of this technique, insights such as clusters or anomalies from one or several combined views can be used to influence the layout or reordering of the other views. Moreover, the importance of nodes and node groups can be detected, computed, and visualized by considering several layout and reordering properties in combination as well as different edge properties for the same set of nodes. As an additional feature set, an automatic identification of groups, clusters, and outliers is provided over time, and based on the visual outcome of the node-link and matrix visualizations, the repertoire of the supported layout and matrix reordering techniques is extended, and more interaction techniques are provided when considering the dynamics of the graph data. Finally, a small user experiment was conducted to investigate the usability of the proposed approach. The usefulness of the proposed tool is illustrated by applying it to a graph dataset, such as e co-authorships, co-citations, and a Comprehensible Perl Archive Network distribution.

Picture Camp

‘Picture Camp’ provides a review of Kirby handle calculus for describing 4-manifolds via decorated link diagrams, as well as techniques for how to simplify such diagrams. This chapter applies these techniques to describe gropes and towers, from the previous chapter, using Kirby diagrams. In addition to decorated links, the diagrams include the information of framings for the attaching and tip regions. In particular, it is shown how to combine two diagrams together when the corresponding spaces are identified along their attaching and tip regions. The chapter also relates the combinatorics of gropes and towers to the combinatorics of the associated link diagrams.

Homophily at a glance: visual homophily estimation in network graphs is robust under time constraints

Network graphs are used for high-stake decision making in medical and other contexts. For instance, graph drawings conveying relatedness can be relevant in the context of spreading diseases. Node-link diagrams can be used to visually assess the degree of homophily in a network—a condition where a presence of the link is more likely when nodes are similar. In an online experiment (N = 531), we tested how robustly laypeople can judge homophily from node-link diagrams and how variation of time constraints and layout of the diagrams affect judgments. The results showed that participants were able to give appropriate judgments. While granting more time led to better performance, the effects were small. Rather, the first seconds account for most of the information an individual can extract from the graphs. Furthermore, we showed a difference in performance between two types of layouts (bipartite and polarized). Results have consequences for communicating the degree of homophily in network graphs to the public.

Writhe-like invariants of alternating links

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are “writhe-like” invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.

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.

The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, which measure depths of regions inside the web. As an application, we resolve an open conjecture that the change of basis between the so-called Specht basis and web basis of this symmetric group representation is unitriangular for $\mathfrak{sl}_3$-webs ([ 33] and [ 29].) We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.

Who talks to whom: an evaluation of a call log visualization

Adding temporal information to social network visualizations is still a challenging task despite previous research efforts. Visualizing call logs on an event-based level can show various attributes of a connection. The dimension time is of great interest to analysts as it offers insights into trends and patterns such as changing relationships between different actors or economic opportunities for businesses. Yet current approaches suffer from limitations that can be improved with the visualization design presented in this work. Our presented visualization was developed considering aesthetic criteria and characteristics of adjacency matrices and node-link diagrams. A heuristic evaluation according to these criteria was conducted. In a formative evaluation process, an artificial dataset was specifically created to examine dynamic social networks. A qualitative user study with observation and think-aloud protocols was conducted and analyzed with regard to the user’s strategies, limitations of the visualization and potential additional features. The visualization appears to be suitable for all of the evaluated network tasks; however, path-related tasks were more challenging than other tasks. Graphical abstract