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To an oriented knot we associate a trace graph in a thickened torus in such a way that knots are isotopic if and only if their trace graphs can be connected by moves of finitely many standard types. For closed braids with a fixed number of strands, we recognize trace graphs up to equivalence excluding one type of moves in polynomial time with respect to the… (More)

- V. KURLIN
- 2007

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… (More)

- V. KURLIN
- 2004

For each n ≥ 2, we construct a finitely presented semigroup RSG n. The center of RSG n 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 three-page 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)

- V. KURLIN
- 2006

The classical Baker-Campbell-Hausdorff formula gives a recursive way to compute the Hausdorff series H = ln(e X e Y) for non-commuting X, Y. Formally H lives in the graded completion of the free Lie algebra L generated by X, Y. We present a closed explicit formula for H = ln(e X e Y) in a linear basis of the graded completion of the free metabelian Lie… (More)

- Vitaliy Kurlin
- 2014 IEEE Conference on Computer Vision and…
- 2014

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)

- V Kurlin, V Vershinin
- 2008

Construction of a semigroup with 15 generators and 84 relations is given. The center of this semigroup is in one-to-one correspondence with the set of all isotopy classes of non-oriented singular knots (links with finitely many double intersections in general position) in R 3 .

- DRINFELD ASSOCIATORS, V. KURLIN
- 2004

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)

- Vitaliy Kurlin
- Comput. Graph. Forum
- 2015

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)

- 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)

- V. KURLIN
- 2009

We design an algorithm writing down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and quadratic in the number of robots. The computing algorithm implies that… (More)