Noncommutative Distance Theory Note 5 Kontsevich Invariant / January 3, 2008

Noncommutative Distance Theory

Note 5

Kontsevich Invariant

TANAKA Akio

R³ : = C × R

Knot     K

Parameter of height     t

Two points on K at t     z ( t )  z’ ( t )

Selected point of z and z’     P

z and z’ Code figure on S¹     D_P

Iteration integral     Z’ ( K ) : = Σ_m=0^∞  ×(-1)^#P1D_P

Quotient vector space that is quoted by 3 relations ( AS, IHX and STU )* over C on which Jacobi figure is described A ( S¹ )

Kontsevich invariant    Z ( K ) ∈ A ( S¹ )

[Note]

*3 relations ( AS, IHX and STU ) are seemed to be related with characters’ descriptive system.

Tokyo January 3, 2008

Sekinan Research Field of Language