Narrow Results

For a knot or link K, let L(K) denote the rope length of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well-known that there exist positive constants c₁, c₂ such that for any knot or link K, c₁ · (Cr(K))^3/4 ≤ L(K) ≤ c₂ · (Cr(K))^3/2. It is also known that for any real number p such that 3/4 ≤ p ≤ 1, there exists a family of knots {K_n} with the property that Cr(K_n) → ∞ (as n → ∞) such that L(K_n) = O(Cr(K_n)^p). However, it is still an open question whether there exists a family of knots {K_n} with the property that Cr(K_n) → ∞ (as n → ∞) such that L(K_n) = O(Cr(K_n)^p) for some 1 < p ≤ 3/2. In this paper, we show that there are many families of prime alternating Conway algebraic knots {K_n} with the property that Cr(K_n) → ∞ (as n → ∞) such that L(K_n) can grow no faster than linearly with respect to Cr(K_n).

We use the new approach of braiding sequences to prove exponential upper bounds for the number of Vassiliev invariants on knots with bounded braid index, bounded bridge number and arborescent knots.

We prove, that any Vassiliev invariant of degree k is determined by its values on knots with braid index at most k + 1.