We introduce and study the structure of an algebra in the linear space spanned by all regular 3-valent graphs with a prescribed order of edges at every vertex, modulo certain relations. The role of this object in various areas of low dimensional topology is discussed. 0 Introduction Regular graphs of degree 3, i. e. graphs in which every vertex is incident with exactly three edges, often occur in mathematics. Apart from graph theory proper, where such graphs are referred to as `cubic', they appear in a natural way in the topology of 3-manifolds, in the Vassiliev knot invariant theory and in connection with the four colour theorem. It turns out that in all these applications 3-valent graphs are endowed with a natural structure that consists in fixing, at every vertex of the graph, one of the two possible cyclic orders in the set of three edges issuing from this vertex. In the theory of Vassiliev invariants 3-valent graphs can be viewed as elements of the primitive subspace P of th...
|
170
|
Every planar map is four colorable
– Appel, Haken
- 1989
|
|
70
|
The four colour theorem
– Robertson, Sanders, et al.
- 1997
|
|
60
|
Cohomology of knot spaces, Theory of Singularities and its Applications
– Vassiliev
- 1990
|
|
50
|
On the Vassiliev knot invariants
– Bar-Natan
- 1995
|
|
32
|
Finite type invariants of integral homology 3-spheres, Jour. of Knot Theory and its Ramifications 5(1
– Ohtsuki
- 1996
|
|
31
|
Feynman diagrams and lowdimensional topology, First European
– Kontsevich
- 1994
|
|
26
|
A calculus for framed links in
– Kirby
- 1978
|
|
23
|
An invariant of integral homology 3spheres which is universal for all finite type invariants
– Le
|
|
20
|
A representation of orientable combinatorial 3-manifolds
– Lickorish
- 1962
|
|
18
|
On the geographical problem of the four colours
– Kempe
|
|
18
|
Remarks on the colouring of maps
– Tait
|
|
16
|
Lower bounds for the number of Vassiliev knot invariants, Topolo
– Chmutov, Duzhin
- 1999
|
|
16
|
The four-color problem, assaults and conquest
– Saaty, Kainen
- 1986
|
|
12
|
The number of primitive Vassiliev invariants up to degree twelve
– Kneissler
- 1997
|
|
11
|
Map coloring and the vector cross product
– Kauffman
- 1990
|
|
10
|
An algebraic approach to the planar coloring problem
– Kauffman, Saleur
- 1993
|
|
9
|
Modifications and cobounding manifolds
– Wallace
- 1960
|
|
7
|
A polynomial invariant of integral homology 3-spheres
– Ohtsuki
- 1995
|
|
6
|
Remarks on the Vassiliev knot invariants coming from sl 2 , Topology 36
– Chmutov, Varchenko
- 1997
|
|
6
|
Applications of negative dimensional tensors, In: Combinatorial mathematics and its applications
– Penrose
- 1971
|
|
5
|
Melvin-Morton conjecture and primitive Feynman diagrams
– Vaintrob
- 1996
|
|
3
|
1890) Map-colour theorem. Quarterly
– Heawood
- 1976
|
|
3
|
Map Coloring, q-Deformed Spin Networks, and Turaev-Viro Invariants for 3-Manifolds
– Kauffman
- 1991
|
|
3
|
Algorithmic and computer methods in three-dimensional topology, (in Russian; English transl.: Algorithmic and computer methods for three-manifolds
– Fomenko, Matveev
- 1997
|
|
2
|
The Arhus invariant of rational homology 3-spheres I: A highly non trivial flat connection on S , preprint
– Bar-Natan, Garoufalidis, et al.
- 1997
|
|
2
|
Vassiliev knot invariants I. Introduction
– Chmutov, Duzhin, et al.
- 1994
|
|
2
|
Is the four color theorem true
– Kainen
- 1993
|
|
2
|
Progress on the four-color theorem
– Seymour
- 1994
|
|
1
|
Lie algebras and the four color theorem, Arhus University preprint
– Bar-Natan
- 1995
|
|
1
|
On the colouring of maps, Proc
– Cayley
|
|
1
|
The Kontsevich integral. Preprint of Max Planck Institut fur Mathematik
– Chmutov, Duzhin
- 1997
|
|
1
|
WWW-page http://www.cs.uwa.edu.au/~gordon/cubics/ containing tables of 3-graphs
– Royle
- 1996
|
|
1
|
Private communication
– Dasbach
- 1997
|
|
1
|
Programs and data available by anonymous ftp from the server ftp://math.botik.ru/, directory pub/a3g
– Kaishev
|
|
1
|
Reinventing the wheel
– Garoufalidis
- 1997
|
|
1
|
On finite type invariants of 3-manifolds III: Manifold weight systems, preprint
– Garoufalidis, Ohtsuki
- 1997
|
|
1
|
Ohtsuki's invariants are of finite type, preprint of Queen Mary and
– Kricker, Spence
- 1996
|
|
1
|
mathematical olympiad for high-school students
– Russian
- 1995
|
|
1
|
Braids and 3-Manifolds
– Prasolov, Sossinsky, et al.
- 1997
|
|
1
|
Algebraic structures on modules of diagrams, Institut de Math'ematiques de Jussieu, Pr'epublication 32
– Vogel
- 1995
|
