Is there a Hilbert's Nullstellensatz for finite type invariants of links?
July-24-11
12:42 PM
Is there a Hilbert's Nullstellensatz for finite type invariants of links? 
posted June 2001 
Let k be an algebraically closed field and let I be an ideal in the polynomial ring k[x1,...,xn]. The Hilbert Nullstellensatz (see e.g. [E]) says that the ideal of polynomials in k[x1,...,xn] that vanish on the variety defined by the common zeros of all polynomials in I is the radical of I. 
Problem: Is there a similar statement for finite type invariants of links? Let I be an ideal in the algebra V of finite type invariants of links. Let Z be the set of links that are annihilated by all members of I, and let J be the ideal in V of all invariants that vanish on Z. Clearly, J always contains the radical of I. Are they always equal? 
Example: Let I be the ideal generated by linking numbers. In this case, Z is the set of algebraically split links. Is it true that every finite type invariant that vanishes on algebraically split links is a sum of multiples of linking numbers by arbitrary other finite type invariants of links? I believe it is true, and I believe it follows from the results of Appleboim [A], but I'm afraid Appleboim's paper is incomplete and while I believe it I cannot vouch for its validity. 
Remark: One may also ask, "what is the Zariski closure of a given set of links?". I believe that in the light of the paragraphs above the meaning of this question should be clear. I know of at least one interesting example: In [N] Ng shows that the Zariski closure of the set of ribbon knots is the set of knots whose Arf invariant vanishes. 
References: 
[A]
E. Appleboim, Finite type invariants of links with fixed linking matrix, arXiv:math.GT/9906138. 
[E]
D. Eisenbud, Commutative Algebra With a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, 1994. 
[N]
K. Y. Ng, Groups of ribbon knots, arXiv:q-alg/9502017. 