scholarly journals Multistring based matrices

2020 ◽  
Vol 29 (06) ◽  
pp. 2050038
Author(s):  
David R. Freund
Keyword(s):  
Homotopy Class ◽  
Open Problem ◽  
Chord Diagram ◽  
Homotopy Classes ◽  
Chord Diagrams ◽  
Text String ◽  
Reidemeister Moves

A virtual[Formula: see text]-string is a chord diagram with [Formula: see text] core circles and a collection of arrows between core circles. We consider virtual [Formula: see text]-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string [Formula: see text], Turaev associated a based matrix that encodes invariants of the virtual homotopy class of [Formula: see text]. We generalize Turaev’s method to associate a multistring based matrix to a virtual [Formula: see text]-string, addressing an open problem of Turaev and constructing similar invariants for virtual homotopy classes of virtual [Formula: see text]-strings.

scholarly journals Genus Ranges of Chord Diagrams

2015 ◽  
Vol 24 (04) ◽  
pp. 1550022 ◽  
Author(s):  
Jonathan Burns ◽  
Nataša Jonoska ◽  
Masahico Saito
Keyword(s):  
Outer Boundary ◽  
Orientable Surface ◽  
Fixed Number ◽  
Chord Diagram ◽  
Boundary Circle ◽  
Line Segments ◽  
Chord Diagrams ◽  
Computer Calculations

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can be, and those that cannot be, realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.

The Ninth Homotopy Class of Spatial 3R Serial Regional Manipulators

2006 ◽  
Vol 129 (4) ◽  
pp. 445-448 ◽  
Author(s):  
Davide Paganelli
Keyword(s):  
Homotopy Class ◽  
Classification Method ◽  
Weak Point ◽  
Homotopy Classes ◽  
Important Notion

Singularities form surfaces in the jointspace of a serial manipulator. Paï and Leu (Paï and Leu, 1992, IEEE Trans. Rob. Autom., 8, pp. 545–559) introduced the important notion of generic manipulator, the singularity surfaces of which are smooth and do not intersect with each other. Burdick (Burdick, 1995, J. Mech. Mach. Theor., 30, pp. 71–89) proposed a homotopy-based classification method for generic 3R manipulators. Through this classification method, it was stated in Wenger, 1998, J. Mech. Des., 120, pp. 327–332 that there exist exactly eight classes of generic 3R manipulators. A counterexample to this classification is provided: a generic 3R manipulator belonging to none of the eight classes identified in (Wenger, 1998, J. Mech. Des., 120, pp. 327–332) is presented. The weak point of the proof given in (J. Mech. Des., 120, pp. 327–332) is highlighted. The counterexample proves the existence of at least nine homotopy classes of generic 3R manipulators. The paper points out two peculiar properties of the manipulator proposed as a counterexample, which are not featured by any manipulator belonging to the eight homotopy classes so far discovered. Eventually, it is proven in this paper that at most four branches of the singularity curve can coexist in the jointspace of a generic 3R manipulator and therefore at most eleven homotopy classes are possible.

Suspension of the Lusternik-Schnirelmann Category

1970 ◽  
Vol 22 (6) ◽  
pp. 1129-1132
Author(s):  
William J. Gilbert
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Category Structure ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Cw Complexes ◽  
Lusternik Schnirelmann ◽  
Image Position ◽  
Simply Connected ◽  
Homotopy Invariants

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],We prove thatandIt is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

scholarly journals Typical representatives of free homotopy classes in multi-punctured plane

2019 ◽  
Vol 11 (03) ◽  
pp. 623-659
Author(s):  
Maxim Arnold ◽  
Yuliy Baryshnikov ◽  
Yuriy Mileyko
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Measure ◽  
Homotopy Class ◽  
Piecewise Linear ◽  
Homotopy Classes ◽  
Simple Method ◽  
Monte Carlo Techniques ◽  
Free Homotopy Class

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii’s theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple method for approximating shortest loops.

Some properties of Hopf-type constructions

1995 ◽  
Vol 117 (2) ◽  
pp. 287-301 ◽  
Author(s):  
Martin Arkowitz ◽  
Paul Silberbush
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Smash Product ◽  
Homotopy Classes ◽  
One To One

If f: X × Y → Z is a map, then the classical Hopf construction associates to f a map hf: X * Y → ΣZ, where X * Y is the join of X and Y and ΣZ the suspension of Z. Since X * Y has the homotopy type of Σ(X Λ Y), the suspension of the smash product of X and Y, the homotopy class of hf can be regarded as an element Hf ↦ [Σ(X Λ Y), ΣZ]. Now elements of [Σ(X Λ Y), ] are in one to one correspondence with homotopy classes in the group [σ(X Λ Y), ΣZ] which are trivial on the suspension of the wedge Σ(X ≷ Y).

scholarly journals (1, 2) AND WEAK (1, 3) HOMOTOPIES ON KNOT PROJECTIONS

2013 ◽  
Vol 22 (14) ◽  
pp. 1350085 ◽  
Author(s):  
NOBORU ITO ◽  
YUSUKE TAKIMURA
Keyword(s):  
Equivalence Relation ◽  
Finite Sequence ◽  
Reidemeister Move ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Homotopy Classes ◽  
The Third ◽  
Reidemeister Moves ◽  
Necessary And Sufficient

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).

scholarly journals Ribbon graphs and bialgebra of Lagrangian subspaces

2016 ◽  
Vol 25 (12) ◽  
pp. 1642006 ◽  
Author(s):  
Victor Kleptsyn ◽  
Evgeny Smirnov
Keyword(s):  
Vector Space ◽  
Symplectic Form ◽  
Chord Diagram ◽  
Ribbon Graph ◽  
Dimensional Vector Space ◽  
Ribbon Graphs ◽  
Chord Diagrams ◽  
Intersection Matrix ◽  
Lagrangian Subspace ◽  
Lagrangian Subspaces

To each ribbon graph we assign a so-called [Formula: see text]-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of [Formula: see text]-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of [Formula: see text]-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

scholarly journals THE INTERSECTION GRAPH CONJECTURE FOR LOOP DIAGRAMS

2000 ◽  
Vol 09 (02) ◽  
pp. 187-211 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Equivalence Class ◽  
Intersection Graph ◽  
Chord Diagram ◽  
Single Loop ◽  
Vassiliev Invariants ◽  
Chord Diagrams

The study of Vassiliev invariants for knots can be reduced to the study of the algebra of chord diagrams modulo certain relations (as done by Bar-Natan). Chmutov, Duzhin and Lando defined the idea of the intersection graph of a chord diagram, and conjectured that these graphs determine the equivalence class of the chord diagrams. They proved this conjecture in the case when the intersection graph is a tree. This paper extends their proof to the case when the graph contains a single loop, and determines the size of the subalgebra generated by the associated "loop diagrams." While the conjecture is known to be false in general, the extent to which it fails is still unclear, and this result helps to answer that question.

Integrating a weight system of order n to an invariant of (n−1)-singular knots

1996 ◽  
Vol 05 (01) ◽  
pp. 23-35 ◽  
Author(s):  
MICHEL DOMERGUE ◽  
PAUL DONATO
Keyword(s):  
Weight System ◽  
Combinatorial Formula ◽  
Chord Diagrams ◽  
Reidemeister Moves ◽  
Singular Knots

Starting from a Weight-System denoted by P and defined on the n-Chord-Diagrams with values in an arbitrary Q–module, we give an explicit combinatorial formula for an invariant of (n–1)-singular knots which has P as its derivative. The formula is defined for regular knot projections. Its invariance under singular Reidemeister moves is then proved.

DISTINGUISHING LINK-HOMOTOPY CLASSES BY PRE-PERIPHERAL STRUCTURES

1998 ◽  
Vol 07 (07) ◽  
pp. 925-944 ◽  
Author(s):  
JAMES R. HUGHES
Keyword(s):  
Open Problem ◽  
Group Homomorphism ◽  
Homotopy Classes ◽  
Minimal Weight ◽  
Ambient Isotopy ◽  
Link Homotopy

An open problem in link-homotopy of links in S3 is classification using peripheral invariants, analogous to that of Waldhausen for links up to ambient isotopy. An approach to such a classification was outlined by Levine, but shown not to be feasible by the author. Here, we develop an approach to finding classification counterexamples. The approach requires non-injectivity of a group homomorphism that is completely determined by minimal-weight commutator numbers (equivalent to the first non-vanishing [Formula: see text] invariants of Milnor). For non-injectivity, the minimal-weight commutator numbers must all be non-zero, and satisfy a certain system of polynomials, which we compute for 4- and 5-component links.

Sign in / Sign up

Export Citation Format

Share Document