The Fundamental Theorem of Vassiliev Invariants

2021 ◽  
pp. 101-134
Author(s):  
Dror Bar-Natan ◽  
Alexander Stoimenow
Keyword(s):  
Fundamental Theorem ◽  
Vassiliev Invariants

scholarly journals ON A BASIS FOR THE FRAMED LINK VECTOR SPACE SPANNED BY CHORD DIAGRAMS

2009 ◽  
Vol 18 (12) ◽  
pp. 1663-1680
Author(s):  
BRYAN BISCHOF ◽  
ROMAN KOGAN ◽  
DAVID N. YETTER
Keyword(s):  
Vector Space ◽  
Fundamental Theorem ◽  
Linear Span ◽  
Vassiliev Invariants ◽  
Chord Diagrams ◽  
Framed Link

In view of the result of Kontsevich, now often called "the fundamental theorem of Vassiliev theory", identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the "4T-relation" (and in the unframed case, originally considered by Vassiliev, the "1T-" or "isolated chord relation"), it is a problem of some interest to provide a basis for the space of chord diagrams modulo the 4T-relation. We construct the basis for the vector space spanned by chord diagrams with n chords and m link components, modulo 4T relations for n ≤ 5.

On some definite integrals and a new method of reducing a function of spherical co-ordinates to a series of spherical harmonics

1903 ◽  
Vol 71 (467-476) ◽  
pp. 97-101 ◽  
Keyword(s):  
Spherical Harmonics ◽  
Fundamental Theorem ◽  
New Method ◽  
Definite Integrals

The expansion of a function f(θ) of an angle θ varying between 0 and π in terms of a series proceeding by the sines of the multiples of θ depends on the fundamental theorem, ∫ π 0 sin pθ sin qθ dθ = 0, where p and q are integer numbers not equal to each other.

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