arXiv subjects

V. Kurlin

Publications and source records attributed to V. Kurlin.

11 recordsLinked to original sources

arXiv2009

Connectivity of Random 1-Dimensional Networks

An important problem in wireless sensor networks is to find the minimal number of randomly deployed sensors making a network connected with a given probability. In practice sensors are often deployed one by one along a trajectory of a vehicle, so it is natural to assume that arbitrary probability density functions of distances between successive sensors in a segment are given. The paper computes the probability of connectivity and coverage of 1-dimensional networks and gives estimates for a minimal number of sensors for important distributions.

cs.IT↗

arXiv2009

Computing braid groups of graphs with applications to robot motion planning

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 2-point braid groups of all light planar graphs have presentations where all relators are commutators.

math.GT↗

arXiv2008

A 1-parameter approach to links in a solid torus

To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a study of codimension~2 singularities of link diagrams. For closed braids with a fixed number of strands, trace graphs can be recognized up to equivalence excluding one type of moves in polynomial time with respect to the braid length.

math.GT↗

arXiv2008

Recognizing trace graphs of closed braids

To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral moves in polynomial time with respect to the braid length.

math.GT↗

arXiv2007

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.

math.GT↗

arXiv2007

Fiber quadrisecants in knot isotopies

Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of knots. We give lower bounds for the number of these fiber secants in all isotopies connecting given isotopic knots. The bounds are expressed in terms of invariants calculable in linear time with respect to the number of crossings.

math.GT↗

arXiv2006

The Baker-Campbell-Hausdorff formula in the free metabelian Lie algebra

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 algebra L/[[L,L],[L,L]].

math.QA↗

arXiv2005

Peripherally specified homomorphs of link groups

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.

math.GT↗

arXiv2004

Compressed Drinfeld associators

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 algbera L generated by the symbols a,b,c modulo [a,b]=[b,c]=[c,a]. The main result is a description of compressed associators that satisfy the compressed pentagon and hexagon in the quotient L/[[L,L],[L,L]]. The key ingredient is an explicit form of Campbell-Baker-Hausdorff formula in the case when all commutators commute.

math.GT↗

arXiv2004

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.

math.GT↗

arXiv2003

Three-page embeddings of singular knots

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 three-dimensional space.

math.GT↗