60 research outputs found
Three-page encoding and complexity theory for spatial graphs
We construct a series of finitely presented semigroups. The centers of these
semigroups encode uniquely up to rigid ambient isotopy in 3-space all
non-oriented spatial graphs. This encoding is obtained by using three-page
embeddings of graphs into the product of the line with the cone on three
points. By exploiting three-page embeddings we introduce the notion of the
three-page complexity for spatial graphs. This complexity satisfies the
properties of finiteness and additivity under natural operations.Comment: 32 pages with 9 figures, submitted to J.Knot Theory and Ra
Gauss paragraphs of classical links and a characterization of virtual link groups
A classical link in 3-space can be represented by a Gauss paragraph encoding
a link diagram in a combinatorial way. A Gauss paragraph may code not a
classical link diagram, but a diagram with virtual crossings. We present a
criterion and a linear algorithm detecting whether a Gauss paragraph encodes a
classical link. We describe Wirtinger presentations realizable by virtual link
groups.Comment: 12 pages, 12 figures, v2: new results have been adde
Auto-completion of contours in sketches, maps and sparse 2D images based on topological persistence.
We design a new fast algorithm to automatically complete closed contours in a finite point cloud on the plane. The only input can be a scanned map with almost closed curves, a hand-drawn artistic sketch or any sparse dotted image in 2D without any extra parameters. The output is a hierarchy of closed contours that have a long enough life span (persistence) in a sequence of nested neighborhoods of the input points. We prove theoretical guarantees when, for a given noisy sample of a graph in the plane, the output contours geometrically approximate the original contours in the unknown graph
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