In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant (named after Victor Anatolyevich Vassiliev), is a knot invariant that extends to singular knots with transverse double points, vanishing on those with m + 1 singularities but not on some with m. This concept was developed by Goussarov, Joan Birman, and Xiao-Song Lin. For a knot K, resolving a double point in two ways yields knots K+ and K−, and the invariant satisfies V¹(K′) = V(K+) - V(K−). Such invariants vanish on knots with sufficiently many singularities, defining their finite type order. Analogous finite type invariants exist for 3-manifolds.
Examples
The simplest nontrivial Vassiliev invariant of knots is given by the coefficient of the quadratic term of the Alexander–Conway polynomial. It is an invariant of order two. Modulo two, it is equal to the Arf invariant.
Any coefficient of the Kontsevich invariant is a finite type invariant.
The Milnor invariants are finite type invariants of string links.1
Invariants representation
Michael Polyak and Oleg Viro gave a description of the first nontrivial invariants of orders 2 and 3 by means of Gauss diagram representations. Mikhail N. Goussarov has proved that all Vassiliev invariants can be represented that way.
The universal Vassiliev invariant
In 1993, Maxim Kontsevich proved the following important theorem about Vassiliev invariants: For every knot one can compute an integral, now called the Kontsevich integral, which is a universal Vassiliev invariant, meaning that every Vassiliev invariant can be obtained from it by an appropriate evaluation. It is not known at present whether the Kontsevich integral, or the totality of Vassiliev invariants, is a complete knot invariant, or even if it detects the unknot. Computation of the Kontsevich integral, which has values in an algebra of chord diagrams, turns out to be rather difficult and has been done only for a few classes of knots up to now. There is no finite-type invariant of degree less than 11 which distinguishes mutant knots.2
See also
Further reading
- Victor A. Vassiliev, Cohomology of knot spaces. Theory of singularities and its applications, 23–69, Adv. Soviet Math., 1, American Mathematical Society, Providence, RI, 1990.
- Joan Birman and Xiao-Song Lin, Knot polynomials and Vassiliev's invariants. Inventiones Mathematicae, 111, 225–270 (1993)
- Bar-Natan, Dror (1995). "On the Vassiliev knot invariants". Topology. 34 (2): 423–472. CiteSeerX 10.1.1.511.6301. doi:10.1016/0040-9383(95)93237-2.
External links
- Weisstein, Eric W. "Vassiliev Invariant". MathWorld.
- "Finite Type (Vassiliev) Invariants", The Knot Atlas.
References
Habegger, Nathan; Masbaum, Gregor (2000). "The Kontsevich integral and Milnor's invariants". Topology. 39 (6): 1253–1289. doi:10.1016/S0040-9383(99)00041-5. http://people.math.jussieu.fr/~masbaum/K39.ps.gz ↩
Murakami, Jun. "Finite-type invariants detecting the mutant knots" (PDF). http://www.f.waseda.jp/murakami/papers/finitetype.pdf ↩