Vitaliy Kurlin

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A classical link in 3-space can be represented by a Gauss phrase encoding a plane diagram of the link in a purely combinatorial way. A Gauss phrase may give rise not to a classical link, but to a diagram with virtual crossings. We describe a linear algorithm determining whether a Gauss phrase encodes a classical link. We characterize Wirtinger presentations(More)
Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in noisy clouds in the plane. The holes in a given cloud are quantified by the topological persistence of their boundary contours when the cloud is analyzed at all possible scales. We design the algorithm to count holes that are most persistent(More)
  • Vitaliy Kurlin
  • 2014 16th International Symposium on Symbolic and…
  • 2014
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)(More)
Drinfeld associator is a key tool in computing the Kontsevich integral of knots. A Drinfeld associator is a series in two non-commuting variables, satisfying highly complicated algebraic equations — hexagon and pentagon. The logarithm of a Drinfeld associator lives in the Lie algebra L generated by the symbols a, b, c modulo [a, b] = [b, c] = [c, a]. The(More)
Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra parameters such as a scale or a noise bound. We define a homologically persistent skeleton, which depends only(More)
For each n ≥ 2, we construct a finitely presented semigroup RSGn. The center of RSGn encodes uniquely up to rigid ambient isotopy in R 3 all non-oriented spatial graphs with vertices of degree ≤ n. This encoding is obtained by using threepage embeddings of graphs into the product Y = T × I, where T is the cone on three points, and I ≈ [0, 1] is the unit(More)