Structure and Semantics of Arrow Diagrams

2005 ◽  
pp. 232-250 ◽  
Author(s):  
Yohei Kurata ◽  
Max J. Egenhofer
Keyword(s):  
Arrow Diagrams

COMPOSITIONS AND THE CHAIN RULE USING ARROW DIAGRAMS

PRIMUS ◽  
1995 ◽  
Vol 5 (3) ◽  
pp. 291-295 ◽  
Author(s):  
J. B. Thoo
Keyword(s):  
Chain Rule ◽  
Arrow Diagrams

Arrow Diagrams are Line Digraphs

1968 ◽  
Vol 16 (6) ◽  
pp. 1141-1145 ◽  
Author(s):  
Dennis P. Geller ◽  
Frank Harary
Keyword(s):  
Arrow Diagrams

scholarly journals A diagrammatic approach to link invariants of finite degree

2004 ◽  
Vol 94 (2) ◽  
pp. 295 ◽  
Author(s):  
Olof-Petter Östlund
Keyword(s):  
Finite Degree ◽  
Explicit Formulas ◽  
Connected Sum ◽  
Knot Invariants ◽  
Graphical Calculus ◽  
Link Invariants ◽  
Low Degree ◽  
Coalgebra Structure ◽  
Arrow Diagrams

In [5] M. Polyak and O. Viro developed a graphical calculus of diagrammatic formulas for Vassiliev link invariants, and presented several explicit formulas for low degree invariants. M. Goussarov [2] proved that this arrow diagram calculus provides formulas for all Vassiliev knot invariants. The original note [5] contained no proofs, and it also contained some minor inaccuracies. This paper fills the gap in literature by presenting the material of [5] with all proofs and details, in a self-contained form. Furthermore, a compatible coalgebra structure, related to the connected sum of knots, is introduced on the algebra of based arrow diagrams with one circle.

scholarly journals On Some Moves on Links and the Hopf Crossing Number

2020 ◽  
Vol 18 (1) ◽  
Author(s):  
Maciej Mroczkowski
Keyword(s):  
Primitive Root ◽  
Jones Polynomial ◽  
Crossing Number ◽  
Equivalence Classes ◽  
Lens Spaces ◽  
Root Of Unity ◽  
Arrow Diagrams

AbstractWe consider arrow diagrams of links in $$S^3$$ S 3 and define k-moves on such diagrams, for any $$k\in \mathbb {N}$$ k ∈ N . We study the equivalence classes of links in $$S^3$$ S 3 up to k-moves. For $$k=2$$ k = 2 , we show that any two knots are equivalent, whereas it is not true for links. We show that the Jones polynomial at a k-th primitive root of unity is unchanged by a k-move, when k is odd. It is multiplied by $$-1$$ - 1 , when k is even. It follows that, for any $$k\ge 5$$ k ≥ 5 , there are infinitely many classes of knots modulo k-moves. We use these results to study the Hopf crossing number. In particular, we show that it is unbounded for some families of knots. We also interpret k-moves as some identifications between links in different lens spaces $$L_{p,1}$$ L p , 1 .

Arrow Diagrams—II

1970 ◽  
pp. 49-78
Author(s):  
Albert Battersby
Keyword(s):  
Arrow Diagrams

scholarly journals KONTSEVICH INTEGRAL FOR KNOTS AND VASSILIEV INVARIANTS

2013 ◽  
Vol 28 (17) ◽  
pp. 1330025 ◽  
Author(s):  
P. DUNIN-BARKOWSKI ◽  
A. SLEPTSOV ◽  
A. SMIRNOV
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Knot Theory ◽  
Experimental Results ◽  
Theory Approach ◽  
Quantum Field ◽  
Computational Results ◽  
Vassiliev Invariants ◽  
Temporal Gauge ◽  
Arrow Diagrams

We review quantum field theory approach to the knot theory. Using holomorphic gauge, we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in terms of arrow diagrams. Explicit examples and computational results are presented.

GAUSS DIAGRAM INVARIANTS OF LINKS IN M2 × R1

2011 ◽  
Vol 20 (03) ◽  
pp. 371-387
Author(s):  
S. A. GRISHANOV ◽  
V. A. VASSILIEV
Keyword(s):  
Infinite Series ◽  
Gauss Diagram ◽  
Arrow Diagrams

We construct an infinite series of invariants of Fiedler type (i.e. composed of oriented arrow diagrams arranged by elements of H1(M3)) for multicomponent links in M3 = M2 × R1, M2 orientable with π1(M2) ≠ {1}.

Segment-arrow diagrams and invariants of ornaments

2000 ◽  
Vol 191 (11) ◽  
pp. 1635-1666 ◽  
Author(s):  
A B Merkov
Keyword(s):  
Arrow Diagrams

scholarly journals Arrow diagrams on spherical curves and computations

2021 ◽  
Vol 30 (07) ◽  
Author(s):  
Noboru Ito ◽  
Masashi Takamura
Keyword(s):  
The Other ◽  
Type Formula ◽  
Ambient Isotopy ◽  
Definition Of ◽  
Arrow Diagrams

We give a definition of an integer-valued function [Formula: see text] derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free [Formula: see text]-module generated by the arrow diagrams with at most [Formula: see text] arrows, called relators of Type ([Formula: see text]) (([Formula: see text]), ([Formula: see text]), ([Formula: see text]) or ([Formula: see text]), respectively), and introduce another function [Formula: see text] to obtain [Formula: see text]. One of the main results shows that if [Formula: see text] vanishes on finitely many relators of Type ([Formula: see text]) (([Formula: see text]), ([Formula: see text]), ([Formula: see text]) or ([Formula: see text]), respectively), then [Formula: see text] is invariant under the deformation of type RI (strong RI[Formula: see text]I, weak RI[Formula: see text]I, strong RI[Formula: see text]I[Formula: see text]I or weak RI[Formula: see text]I[Formula: see text]I, respectively). The other main result is that we obtain new functions of arrow diagrams with up to six arrows explicitly. This computation is done with the aid of computers.

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