On Finiteness of Vassiliev Invariants and a Proof of the Lin-wang Conjecture via Braiding Polynomials

Using the new approach of braiding sequences we give a proof of the Lin-Wang conjecture, stating that a Vassiliev invariant v of degree k has a value Ov(c(K)k) on a knot K, where c(K) is the crossing number of K and Ov depends on v only. We extend our method to give a quadratic upper bound in k for the crossing number of alternating/positive knots, the values on which suffice to determine uniquely a Vassiliev invariant of degree k. This also makes orientation and mutation sensitivity of Vassiliev invariants decidable by testing them on alternating/positive knots/mutants only. We give an exponential upper bound for the number of Vassiliev invariants on a special class of closed braids.