Stable Treemaps via Local Moves

Tightening Curves on Surfaces via Local Moves

We prove new upper and lower bounds on the number of homotopy moves required to tighten a closed curve on a compact orientable surface (with or without boundary) as much as possible. First, we prove that Ω(n2)moves are required in the worst case to tighten a contractible closed curve on a surface with non-positive Euler characteristic, where n is the number of self-intersection points. Results ...

Perceiving a stable environment when one moves.

Squarified Treemaps

An extension to the treemap method for the visualization of hierarchical information, such as directory structures and organization structures, is presented. The standard treemap method often gives thin, elongated rectangles. As a result, rectangles are difficult to compare and to select. A new method is presented to generate lay-outs in which the rectangles approximate squares. To strenghten t...

Local Moves and Gordian Complexes, II

By the works of Levine [2] and Rolfsen [5], [6], it is known that a local move called a crossing-change is strongly related to the Alexander invariant. In this note, we will consider to what degree the relationship is strong. Let K be a knot, and K the set of knots obtained from a knot K by a single crossing-change. Let MK be the Alexander invariant of a knot K, and MK the set of the Alexander ...

Stochastic Evolution of Graphs using Local Moves

Inspired by theories such as Loop Quantum Gravity, a class of stochastic graph dynamics was studied in an attempt to gain a better understanding of discrete relational systems under the influence of local dynamics. Unlabeled graphs in a variety of initial configurations were evolved using local rules, similar to Pachner moves, until they reached a size of tens of thousands of vertices. The effe...