Knots and Links without Parallel Tangents

Steinhaus conjectured that every closed oriented C-curve has a pair of anti-parallel tangents. The conjecture is not true. Porter [Po] showed that there exists an unknotted curve which has no anti-parallel tangents. Colin Adams rised the question of whether there exists a nontrivial knot in R which has no parallel or antiparallel tangents. In this paper we will solve this problem, showing that any (smooth or polygonal) link L in R is isotopic to a smooth link L̂ which has no parallel or antiparallel tangents. If S(L) is the set of all smooth links isotopic to L, then the subset L̂(L) of all L̂ which has no parallel or antiparallel tangents is not dense in S(L) if it is endowed with C topology. However, L̂(L) is dense in S(L) under C topology. We will show that any neighborhood of L contains such a link L̂. See Theorem 7 below. The result has some impact on studying supercrossing numbers, see the recent work of Pahk [Pa]. We refer the readers to [Ro] for concepts about knots and links. Throughout this paper, we will use I to denote a closed interval on R. Denote by S the unit sphere in R , and by S1 the circle S ∩Rxy on S , where Rxy denotes the xy-plane in R . Denote by Z[z1, z2] the set {v = (x, y, z) ∈ R 3 | z1 ≤ z ≤ z2}. Similarly for Y [y1,∞) etc. A curve β : I → R is an unknotted curve in Z[z1, z2] if (i) β is a properly embedded arc in Z[z1, z2], with endpoints on different components of ∂Z[z1, z2], and (ii) β is rel ∂ isotopic in Z[z1, z2] to a straight arc. Given a curve α : I = [a, b] → S and a positive function f : I → R+ = {x ∈ R|x > 0}, we use β = β(f, α, t0, v0) to denote the integral curve of fα with β(t0) = v0, where t ∈ I. More explicitly, define