HAMILTONIAN CYCLES AND ROPE LENGTHS OF CONWAY ALGEBRAIC KNOTS

Abstract

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 {Kn} with the property that Cr (Kn) → ∞ ( as n → ∞) such that L(Kn) = O( Cr (Kn)p). However, it is still an open question whether there exists a family of knots {Kn} with the property that Cr (Kn) → ∞ ( as n → ∞) such that L(Kn) = O( Cr (Kn)p) for some 1 < p ≤ 3/2. In this paper, we show that there are many families of prime alternating Conway algebraic knots {Kn} with the property that Cr (Kn) → ∞ ( as n → ∞) such that L(Kn) can grow no faster than linearly with respect to Cr (Kn).